e.

FOR THE PEOPLE FOR EDVCATION FOR SCIENCE

LIBRARY OF THE AMERICAN MUSEUM OF NATURAL HISTORY

°

TRANSACTIONS

AND aT yh &@¢. ao PROCEEDINGS 7

OF THE

Ropal Society of Victoria.

WOE Xx EEL

Edited under the Authority of the Council of the Society.

THE AUTHORS OF THE SEVERAL PAPERS ARE SOLELY RESPONSIBLE FOR THE SOUNDNESS OF THE OPINIONS GIVEN AND FOR THE ACCURACY OF THE STATEMENTS MADE THEREIN.

7 MELBOURNE: ras OoN. PIRTH & M*‘CUTCHEON, PBINTERS: FLINDERS LANE WEST,

ISSUED i1hth MARCH, 1878.

AGENTS TO THE SOCIETY.

WILLIAMS & NORGATE, 14 HENRIETTA STREET, COVENT GARDEN, LONDON.

To whom all communications for transmission to the Royal Society of Victoria from all parts of Europe should be sent,

4

PRHFACH. |

THE publication of Volume XIII. has been unavoidably delayed so long mainly with the idea of printing two years’ transactions in one volume. It has, however, been thought better to issue each year’s transactions separately. Volume XIV. will be ready in a month or two, and in future each year’s transactions will be prepared for issue.

at the following Annual Meeting.

Hopal Society of Victoria. Ese 7) @

| parron. ' HIS EXCELLENCY SIR GEORGE BOWEN, G.C.M.G.

president. R. L. J. ELLERY, Esa., F.R.S.

Gite-Dresidvents. GEORGE FOORD, Esq,, F.C.S. | E. J. WHITE, Esq., F R.A.S8.

How, Grersurer. PERCY DE J. GRUT, Esq.

Hon. Secretary.

F. J. PIRANI, Ese., M.A.

Bon. Librarian. J. E, NEILD, Esq., M.D.

Council. A. C. ALLAN, Ese. S. W. M‘GOWAN, Esa. H. M. ANDREW, Ese., M.A. H. K. RUSDEN, Ese. J. BOSISTO, Esq., M.L.A. THOS. E. RAWLINSON Esa., C.E. W. C. KERNOT, Ese., M.A. JAMES T. RUDALL, Esq, F.B.C.S. PROFESSOR E. J. NANSON. F. POOLMAN, Ese. E, HOWITT, Ese. G. H. F. ULRICH, Ese, F.G.S.

ie aa

Phe

Roval Society of Victoria.

ANNIVERSARY ADDRESS

OF

Che President,

Mr. R. L. J. Every, F.R.A.S., Government Astronomer.

(Delivered to the Members of the Royal Society, at their Annual - Conversazione, held on Thursday, 10th August, 1876.)

Your EXCELLENCY AND GENTLEMEN OF THE RoyYAL SOcrgTY,

The 12th Rule of our Society, relating to the time at which the Presidential Address shall be delivered, has of late years been more honoured in the breach than in the observance ; every year it has gota little later—this year later than ever, and the usual phrase—‘We meet to Inaugurate our session,’ has become inappropriate. I must confess, however, that this bad habit has come into fashion since I have had the honour of being your President ; the remedy, therefore, is obvious. We meet this evening to commemorate the entry of the Society into its 19th session by a social gathering of our members and their friends, as has been our custom for several years past, and the only really formal business of the evening provided for by our rules—the delivery of address—now devolves on me as your President. ,

In doing this I wish to be as brief as possible. Since I had last the honour of addressing you, about two years ago,

xi President's Address

you have done me the honour of twice re-electing me your President ; and now, perhaps, is a fitting occasion to assure you how highly I appreciate the confidence you thus place in me. I have sometimes felt I should like to be relieved of the responsibility and anxiety of the position, and make room for a better man; but as each year has come around I have found myself nominated and re-elected without pro- testing against the honours you heap upon my head. I need scarcely tell you, gentlemen, that I take the greatest interest in the welfare of this Society, and I shall always be ready, as long as I have good health, to do my best for its good and advancement, whatever position | may hold in its ranks.

You will be eA to learn that the financial position of our Society is now better that it has been for some years. Our revenue proper is not much larger than heretofore, but the resumption of the small annual grant from the Govern- ment has enabled your Council to carry on the printing and other work of the Society in a satisfactory manner without getting into debi. We have now on our rolis 122 members, and I am glad to see among our junior members gentlemen who have been educated in the colony, who, from their acquirements and scientific training, | have reason t0 hope will become most useful acquisitions to the Society.

It has been usual for the President to refer in his address to the papers and other matters which have occupied our meetings held since the preceding conversazione ; but, as the Transactions are now published and issued soon after each meeting, I think it will be unnecessary to refer to them on this occasion; suffice it to say that there have been six meetings held since our annual gathering last year, at which ten papers and other communications were contributed, which, in most cases, led to interesting and instructive ; discussions. While on this subject I may mention that I

for the year 1876. xi

found during my late holiday in Europe that Scientific Societies there are subject to the same phenomenon as we, unfortunately, sometimes witness—namely, paucity of attend- ance at some of the ordinary meetings. There, as here, unless the business of the meetings is unusually interesting and sensational, a few only of the more earnest members attend; and I have been present at several meetings of some of the highest and oldest societies in London where the attendance has been no better than it is in this hall. Small attendances must not, however, be taken asany sign of the want of vitality, for the real functions of this and similar societies are but exhibited in the encouragement and inducement they afford to investigation and experiment, and in the resulting permanent knowledge embodied in their transactions. The small attendance at some of our ordinary meetings, when the business has been of less immediate interest, has induced the Council to arrange that some of them should be of a less formal and more of a conversational character, at which exhibits of new apparatus, intelligence of scientific or other progress, accounts of experiments or observations, not necessarily original, had been received and discussed ; and this plan, so far as has heen tried, has been found ee HOry:-

‘I believe the functions of this Society might possibly be extended with advantage in the direction of brief special lectures for the demonstration of new or interesting facts in physical or other science. Such a course has already been thought of, and | believe is well worthy of putting into practice.

The books in the library have now been thoroughly arranged and classified, and the binding of the periodicals has been commenced, and will be proceeded with from time to time. As regards our publications, I may state that

Volume XI. has been published and issued, and that all B 2

Xiv President's Address

the earlier papers contributed during cur present session are printed and distributed, and the rest in the printer's hands. |

The building and grounds of the Society are in a much better condition than has been the case for some years past, The repairs to the fencing, and the growth of the trees, with the periodical attention given to the ground generally, have | much improved the aspect of affairs. The interior of the building is in a good state of repair, but the appearance of the exterior is exceedingly unsightly. The necessity of getting it stuccoed has been constantly under the notice of the Council, but hitherto the state of the finances has not been such as to warrant it in making the necessary expendi- ture, more especially as they had the assurance of the architect that the building would not suffer for want of stuccoing for some time to come. The Council are of opinion, however, that if for no other reason than appear- ance sake, it is highly desirable to get this work done as soon as the funds will admit.

Leaving the more domestic affairs of the Society, I wish now to call your attention to some of the noteworthy facts connected with the past year’s history of scientific progress.

In Astronomy there appears little of more than passing interest to arrest our attention; it almost seems as if a lull had fallen on this department of science after the unusual activity caused by the transit of Venus in December, 1874. This is apparent only, for nearly all the national observa- tories have been busily engaged, each in its own particular direction. This is true also as regards our own observatory, for while I have nothing sensational to refer to, our principal work—determination of the positions of stars, and the revision of Sir John Herschel’s nebulz with the great tele- scope—has gone on without intermission. Our great tele- scope has new rivals vying with it in probing the great

PaaS

for the year 1876.. XV

depths of the universe. At the Paris observatory a large Newtonian reflector (almost of exactly similar dimensions to our Cassegrainian) has been lately completed, and is now at work ; at Washington the great refractor of 26 inches aperture and 31 feet focal length is actively employed, and in some trials on nebular observation has proved itself no insig-

nificant rival to the large apertures of our and the other

three large reflectors; and further, the maker of the Mel- bourne telescope is now engaged in the construction of another enormous refractor for the Vienna Observatory, which is to be 27 inches aperture and about 33 feet focal length. Now that it is likely there will be more busy eyes and large telescopes occupied on the fainter celestial objects, to the observation of which our reflector has been principally devoted, it becomes all the more necessary that what has already been accomplished here should become known. At present very little of the results of the work of the great telescope has been published. I am now, however, in hopes that this will soon be done, as a method of doing it has been decided upon, and the only cause of delay now is the want of means. ‘This, I have no reasonable doubt, will shortly be forthcoming, when a good account will be given of how this magnificent instrument has been employed since its erection. The final results of the observations of the transit of Venus have not yet been obtained; the laborious calculations involved will probably delay it for some time longer. It is believed, however, from approximate results already arrived at, that the sun’s distance, from these observations, will be

found to be somewhere between the distance obtained by

the transit of Venus in 1769 (corrected by Stone), and the distance obtained by the parallax of Mars in 1862; that is, somewhere between 91,580,000 and 91,240,000 miles. The number of the planetoids (the small planets which occupy the gap between the orbits of Mars and Jupiter) already discovered is 161. Most of these bodies are so minute that

XVI President's Address

their detection among the myriads of small stars is a matter of considerable difficulty, even to accomplished observers ; but, nevertheless, a systematie search for new members of this group with telescopes of adequate power, appears to be always rewarded by discovery. The “Lunar tables,” as they are called, are a series of numbers representing the position, distance, &c., of the moon from day to day or hour to hour calculated for some years in advance, and are of the utmost importance in practical astronomy, navigation, and determination of geographical position generally. It is, however, a remarkable fact that all tables hitherto computed become erroneous after the lapse of years, so that the places given no longer represent the moon’s actual position, and this - would seem at first sight all the more remarkable because her position is and has been continually observed by nearly all the principal national observatories. But the complexity of influences to which she is subjected in her motion through space, coupled with the fact that her mass is probably phy- sically unsymmetrical, makes it an extremely difficult pro- blem to form a theory, taking all these disturbing influences into account, so that tables founded on it shall give the moon's precise position at very distant dates. The tables in the American. Nautical Almanac of Professor Pierce seem however, to be the best yet computed. The veteran Astro- nomer Royal of England, Sir George Airy, who is now in his seventy-sixth year, has lately undertaken to work out a new lunar theory to replace those which experience has shown to be insufficient. He reports that his task is well advanced towards completion, and I am sure all scientific men at least will wish him health and vigour to complete this great self-imposed task for the good of the whole civi- lised world.

In Physical Science also there is nothing of more than ordinary interest to refer to. Mr. Crookes’ investigations on the action of light and heat on bodies in vacuo have been

for the year 1876. XVii

interesting in the highest degree, and although the supposi- tion that the remarkable phenomena exhibited indicated the existence of a new force, which was at first entertained by some, has not been sustained by further investigation and experiment, his researches in this direction have, at least, ‘opened up a new and interesting, if not useful field, in phy- sical science. Concerning this, Mr. Foord will probably have a few words to say in the course of the evening, more especi- ally in reference to a very interesting little apparatus known as Crookes’ radiometer. Some little sensation has been excited lately by the supposed discovery of a new force, allied to electricity, and called etheric force. Some peculiar phenomena, observed with respect to induced electric cur- rents, have been the origin of this supposition. There can be no doubt, however, that they are simply induction phe- nomena, perhaps not hitherto thoroughly investigated, although certainly known, but which with the present ten- dency to discover new forces have been precipitately put in that category.

Although the science of Chemistry svealeas steadily from year to year, it is not quite always that discoveries of popular interest are included among its newer acquisitions; the newly-discovered metal “gallium” is, however, sufficiently remarkable to demand a brief notice on this occasion. For- merly, the processes of humid analysis, including electrolysis, were the only means available for the discovery of new elementary substances ; of late years the much more deli- cate and searching method of spectrum analysis has enabled us to discover—first, rubidium,and cesiwm, then thalliwm, _ afterwards indiwm, and now by its means galliwm has been recognised, and has since been separated. All these are elements ; they are all metals, each possessing definite chemical and other properties. Gallium was discovered in August, 1875, by M. Lecog Boisbanbrau while examining

XVili President's Address

with the spectroscope a blende (a sulphide of zinc) from a mine in the Pyrenees. He observed new and hitherto unrecognised lines in the spectrum, which have enabled him | to pursue, and eventually to separate, and obtain specimens of, the new metal. The chemical and physical properties of this new substance are in some measure ascertained now. __ that the metal has become tangible; but the delicacy of the means by which this has been brought about may be estimated from the statement that the earliest experiment in which the nature of the spectrum of this new metal was established was made on a quantity something less than the 15,000th part of a grain, dissolved in a very small drop of liquid. The melting point of pure gallium is stated to be so low as to warrant our regarding it as being with mer- cury, in the category of metals, fluid at ordinary atmospheric temperatures ; nor are its already ascertained chemical rela- tions less interesting. It has been shown that elementary bodies may be arranged, according to their combining equi- valents, into groups of three, or “triads,” in which the combining equivalent of the middle element is the numerical mean of the two others, but in more than one of these groups the middle term is wanting. From what has been ascer- tained concerning gallium, it appears highly probable that it will be found to fill one of these gaps—that, namely, between aluminium and indium ; and it has been moreover suggested that a wanting element with a combining equivalent, the mean of these of silicon and tin, should be sought in the field of natural combinations respectively of arsenic and titanium. These foreshadowings of the existence of elements new to science of definite characters and positions in the great chemical scheme suggest a comparison with discoveries in another domain of human knowledge—with those, namely, which predicted and led to the discovery of the plane

Neptune.

Ok i a.

for the year 1876. X1x

Some very interesting discussions on the efficacy of the intravenous injection of ammonia in cases of bites by Australian snakes have recently taken place at the Medical Society of Victoria, and perhaps there is no other subject that has cropped up in medical and surgical science during the past year which will have more interest for Australians than this. It, cannot be said. that the result of these discussions, or of the experiments which led to them, is altogether satisfactory, although there can be no doubt that in the evidence.adduced, and the exchange of Opinion, the knowledge of the whole question has been con-

' siderably advanced. When Professor Halford proved that a

powerful agent like ammonia could, under certain conditions, be passed directly into the circulating blood, and so carried mechanically to the heart, and probably the nervous centres, without ‘much danger, and that its effect in his hands ap- peared to be that animals apparently dying, from snake bites especially, were rapidly re-vitalised as it were, it naturally occurred to him as an appropriate remedy to try on the snake-poisoned human subject. This, as you know, was done, and the patient recovered ; many other cases of a more or less similar nature occurred, where recovery from what at the time seemed a hopeless condition was apparently brought about by the injection of ammonia; and the opinion of a large number of intelligent medical men was in favour of the adoption of this treatment for such cases. Other equally intelligent medical men had doubts of the efficacy of this remedy, and eventually a committee of the Medical Society was appointed to carry out a series of experiments to test the value of ammonia injection in snake poison. Their

report was so utterly adverse to the ordinarily received

opinion, that a very animated and interesting discussion took place at several meetings of the Society, but the balance of opinion was still in favour of ammonia injection as a remedy

—— SS ——— —————— ee SEE eEos3EY—e=—=eEeee ee

XxX President's Address

under certain conditions, and I have no doubt it will still be resorted to in nearly every case of snake bite where the life seems in imminent danger. The question naturally sug- gests itself in every case that survives after the treat- ment by ammonia, “Would death have occurred without it?” This, of course, cannot be proved; but the same may be said of all remedies used in medicine or surgery. There can be no doubt, from what transpired at these discussions, that in many cases treated with ammonia the patient was poisoned with alcohol; but who shall say whether the snake or alcohol poison was killing? and if ammonia will save from both, so much the better. However important the intravenous injection of ammonia may be considered in the treatment of snake poison, I think its value as a therapeutic agent in other cases of endangered human life, as shown by some of the collateral evidence given in the discussions referred to, gives broader significance to the whole question than was apparently involved in the late experiments and controversy; and it is to be hoped that both Professor Halford and other of our medical men will extend their investigations and experiments, not only with the view to obtain a more precise idea of the modus operandi of this and other agents introduced directly into the circulation, but also with the view of thoroughly testing the value of this method of applying remedies in urgent cases. The com- paratively modern method of endermic injection has become an inestimable blessing to suffering humanity, and enables the physician and surgeon to confidently use remedies which, administered in the ordinary way to enter the system by digestion, often only afforded relief at the expense of after- exhaustion of vital powers. If, therefore, further investiga-— tion should prove that the intravenous injection of remedies can be as safely and as advantageously used in some cases as the hypodermic injection is in others, it will constitute

for the year 1876. Sor

one of the most important steps in medical science achieved in modern times.

Tn connexion with this subject there is a matter which is exciting some considerable attention in England justnow—I refer to the movement against vivisection. It is,of course, well known that experiments on living animals are frequently made by physiologists and others with the view of extending our knowledge of the vital functions of anatomy, and the action of chemical and other substances, in all cases ostensibly for the benefit of the human race. Of late years, however, a popular belief has grown up among a certain class in England that vivisection and torture of animals was practised to a very large extent in that country without adequate reason, and by persons not influenced by the highest motives, and very strenuous efforts were made to put a stop to such practices. The general public, however, are now convinced that this belief was erroneous in a great measure, and the statements as to the prevalence of the practice exaggerated ; for while well-known and eminent physiologists did resort to vivisection in prosecuting their investigations, it was nearly always with that regard for the suffering or life of God’s creatures which must necessarily influence all truly scientific men. The amount of vivisection practised was very small, and cases of wanton cruelty or needless experiment were found to be exceedingly few. While repudiating any sympathy with that indiscriminate sentimentality which characterised the more violent part of

_ this movement, I am of opinion that some legislation on the matter is highly desirable to protect the earnest investigator

on the one hand from the undue interference of sentimental busybodies, and to prevent an unnecessary resort to vivi- section or experiment on animals, or carelessness or cruelty

in the practice of it when necessary on the other. There

has been a Royal Commission, which has inquired into the

XXli President's Addvess

subject, and Lord Carnarvon has introduced a bill into the British Parliament, which, I think, will be hailed by all right thinking men as a just and righteous provision. The provisions of the bill are categorically given in Nature, and are as follow :—“1. Experiments must be performed with a view only to the advancement, by new discovery, of know- ledge which will be useful for saving or prolonging human life, or alleviating human suffering ; 2, In a registered place; 3, By a person holding a licence from one of .Her Majesty’s principal Secretaries of State; 4, The animal must, during the whole experiment, be under the complete influence of some anesthetic, not urari; and 5, Must be killed before it recovers from the influence of the anzesthetic; 6, The experiment shall not be performed for demonstrational pur- poses; 7, Nor for the purpose of attaining manual skili.” © In former addresses I have on several occasions alluded to the subject of Meteorology somewhat at length, and have, I trust, kept you au courant with the most important points ' in connexion with the advancement of this: branch of knowledge. To us in Australia the value of a better knowledge of the laws that govern the weather can scarcely be overrated, as our prosperity depends so largely on the amount and period of rainfall. Not that it is possible, by any amount of knowledge, to largely modify our climate ; it may become, nevertheless, possible, by systematic investi- gation, to foresee the approach of great disturbances of the atmosphere, or even critical seasons, and to be forewarned is to be forearmed. Ido not think we have data extended over sufficient period or area in Australia to enable any one to safely make any deductions yet. I believe, however, that with the data we already possess, aided by a system of observations over as much of the coast-line as possible, combined with others at representative localities in the interior, and especially in those parts under the influence of

for the year 1876. XXiil

the monsoons, we should be able to ascertain some of the more general laws which govern the weather in Australia, and which will go a long way to help towards the chief desideratum—obtaining a forewarning of storms, and even critical periods and seasons. ‘To this end I have lately invited the co-operation of the directors of Australian observatories in establishing a uniform system of inter- colonial weather telegraphy, which I hope will be in full operation before our next conversazione. In America a most complete system has been in operation for some years, which I described to you on a former occasion. This system has been most successful, and it is stated that 80 per cent. of the predictions—which are published nearly every day—for the several districts over which the observations extend, turn out to be correct. These predictions, however, only refer to the weather from day to day, and not to any lengthened period; but even with this limitation it becomes of immense practical value, and no doubt commensurate with the very large national expenditure which is devoted to it.

A movement has lately been, made in England which promises to be of the utmost importance not only simply as regards science, but also in an educational aspect. I refer to the loan collection of scientific apparatus which has been collected at the museums at Kensington, the public exhibition of which was privately opened by the Queen on May 13th. The proposition for this collection originated in England, where it was made to the Lords of the Committee of the Council on Education, was approved, and assumed a definite shape through the efforts of a committee including over 130 names of the most distinguished men of science. Although the dis- play is in London, the movement is essentially international. Belgium, France, Germany, Italy, the Netherlands, Norway, Russia, Austria and Hungary, Spain, and United States,

XXIV President's Address

have undertaken to contribute, and have opened their museums and scientific storehouses in order that the collec- tion shall be as complete as possible. Whatever intellectual pursuit is aided by instrumental means will be duly repre- sented in this collection ; and there will be brought together not only the instruments of research used at the present time, but many invaluable specimens of the tools with which the early pioneers of human knowledge first began-to ques- tion Nature. The Astrolabe of Tycho Brahe, the telescope of Galileo, will be seen together with the magnificent astronomical instruments of the present day, prominent among which are models of the great Melbourne reflector and the gigantic Vienna refractor of 27 inchesaperture. The various sections are so arranged that in many cases the history of the progress in the respective sciences is more plainly shown than could be done by a written book ; while throughout can be contrasted specimens of the earliest apparatus used in any branch of science with the refined appliances of the present day—Newton’s simple optical apparatus with the exquisite prisms and spectroscopes of to- day ; Dalton’s crude balance with the magnificent weighing- machines of the present time, with the unimpeachable weights of pure quartz. It would occupy too much time to speak of this subject with any justice to its importance. The value, however, of this movement cannot be over-estimated, although—as science as yet unfortunately only interests the few—it may not be so universally appreciated as we could hope. The Times, in an article on the opening of this exhibition, says :—“ The exhibition which Her Majesty the Queen privately visits and opens to-day 1s one of which not only England, but Europe, may he justly proud. Pride, however, is not the only sentiment we English should feel ; for at last, if even only for a brief space, we have, under the name of a loan collection of scientific apparatus, a Science

Pees ae

for the year 1876. XXV

Museum as complete as those in which we have already enshrined our art and literature. For at least six months therefore we shall not only be as rich in this respect as France, Germany, Italy, Holland, and Switzerland, but far richer, since those nations, with an enthusiasm and goodwill which command our universal gratitude, have spoiled their ancient treasure-houses, their laboratories, and private col- - lections, in order that science may be worthily represented among us now that our Government has consented to pro- vide a home, however temporary, for her.”

In conclusion, J would return for a few moments to the immediate affairs of our Society.

I have already referred to the smallness of attendance at some of our ordinary meetings, and to certain propositions for the improvement of the working of the Society. I would, however, exhort our scientific and literary members, and more especially our younger ones, to renewed activity. It cannot be supposed in a small community like ours that enough scientific workers in original investigations can be found to keep this Society in active operation with entirely new matter; but if our legitimate functions be fully exer- cised I can see no reason why we should not have busy sessions and full meetings. The fields of investigation are only too numerous; the further we advance in knowledge the wider they become ; the more science contributes to the welfare, convenience, or luxury of the community, the more is demanded of it. So our young scientists have no lack of scope for their inquiries.

It should be clearly understood that accounts and results

of experiments, the discovery or improvement of mechanical appliances, suggestions of new modes of investigation or

observation,simple observations in natural history, astronomy, chemistry, physiology, medicine, or surgery, besides matters pertaining to the advancement of literature and art, all ° come within the proper province of this Society.

XXvi President's Address for the yewr 1876.

There is surely, then, enough to do. I have often found that most interesting and valuable information has been withheld because of a fear that it was of too trivial a nature, not original, or not sufficiently scientific. It is easier to make mistakes in this direction than in the opposite, for as a rule the Council will always exercise its discretion for the exclusion of contributions manifestly unworthy the attention of the members. If we each do our best for the advancement of knowledge we shall all do something, and J am sure the result will redound to the credit of this Society, as well as of the country we now belong to,

Art. I—On Practical Geodesy. By MARTIN GARDINER, C.E.

[Read 11th May, 1876.]

THE method of investigation employed in this paper is of a purely elementary character, and in this respect it differs from that usually adcpted by the most distinguished geometers who have written on the subject. The method introduced by Legendre, Delambre, and Puissant, and which has been followed by Airy and others, is characterised chiefly by the subsidiary use of the higher calculus and interminable series.

The elementary method here pursued leads to simpler and more comprehensive formulz, and at the same time affords a clearer insight into the various relations between latitudes, azimuths, differences of longitude, length and circular measure of geodesic arc, angles of depression of the chord, &c. Its power of improving and extending the science in one of its most useful directions can be judged of from the numerous new results arrived at, and a com- parison between them and those hitherto evolved by means of the higher calculus.

The errors which have been shewn to exist in some of the investigations and formule given in the “account” of the principal triangulation of Great Britain and Ireland, will no doubt attract the attention of Engineers and Surveyors engaged on trigonometrical surveys in India and elsewhere.

Let P, be the pole of reference of the spheroidal earth ; “3 e. be the centre of the earth ; S,, S,.. be any two stations on the earth’s surface ; wh ra be the points in which the normals at the respective stations 8,, S.,, cut the earth’s polar axis. The planes §,Z Ss 8. 4, are “the normal-chordal planes.” And any ‘plane whatever which contains the: chord B

2 On Practical Geodesy.

of the geodesic arc §,S,, shall be referred to as a chordal plane.

The polar and equatorial radii of the earth being 20855233, and 20926348 feet, it is easy to show that for arcs on its surface not more than 528000 feet or 100 miles in length, we may consider the traces of the two normal-chordal planes as equals in length and circular measure to that of the “true geodesic” or shortest arc between the stations.

Conceive two unit spheres described, having S.,, S,,, as centres. Let C,S, 1 P, be the points in which the sphere S, is pierced by the productions of the lines C.S,, Z,S.,8,S., through the centre S.,and by the line §.P parallel to and in. the same direction as the polar axis C_P..

Let C,,S,, 1, P,, be the points in which the sphere 8,, is

“) MW ME W?

pierced by the productions of the lines CS,,, Z,.S,,, by the chord §,8,, taken in the direction 8,.S,, and by the line S,.£, parallel to and in the same direction as the polar radius C,P..

Then evidently the points P, C,S,, are in the trace, on the unit sphere S., of the earth’s meridian plane through S,; and P,C,,8,, are in the trace, on the unit sphere 58.,, of the earth’s meridian plane through the station §,,.

The arc PI is equal to the arc PI, each of them being the measure of the angle which the chord joining the sta- tions makes with the earth’s polar axis.

The angle PS I is the azimuth of the station S, as observed at the station S,,; and the angle PSI is the sup- plement of the azimuth of the station 8,, as observed at the station S. The ares PS, PS, are the geographic colatitudes of the stations S.S8,.,—such as can be measured directly by means of the Zenith Sector.

The arcs PC, PC, are the geocentric colatitudes of the stations.

Now conceive the unit sphere S,, moved by direct trans- lation along the chord, carrying its lines and points rigidly fixed, until its centre coincides with the centre 8, of the unit sphere 8.. It is evident that the points 1, P_, will coincide witb I, P, and that the points I, C, C,, lie in one great circle of the sphere 8... It is also evident that the points P,S,,C., lie in one great circle of the unit sphere S,, and that the spherical angle SPS, or CPC, is equivalent to the difference of longitude of the stations §,S,..

Let p, p,, be the points in which the lines PS,, P'S.,,., parallel to the polar axis, pierce the earth’s equator. Then

On Practical Geodesy. 3

it is evident that the plane angle p,C,p, is equivalent to the difference of longitude of the stations,

It is also evident that the plane angles C,pp,, C.p,p, are equals respectively to the spherical angle S PI, and the sup- plement of the spherical angle S PI. 3

Let D, D,, be the points in which the great circles IS , IS, cut the great circles PSC, PSC, respectively. It is evident the arc SS, is the measure of the angle which the normals make with each other.

The arc SD, is the measure of the plane angle S,Z,S,, ; the arc 8 Dis the measure of the plane angle 8,,Z,,S,; the ares SC, 8,C,, are the measures of “the angles of the vertical” at the stations $,S,.; the spherical angle SIS, is equal to the angle between the two normal-chordal planes.

And if O, E, E,, be the points in which the great circle of the unit sphere having I as pole cuts the ares SS, SD, SD, respectively ; it is evident that the arcs SE, SE, are the measures of the angles of depression of the geodesic chord S.8,, below the tangent planes to the spheroidal earth at the respective stations S.S,,; and they are the complements of the angles which the normals make with the chord.

The spherical angles SSD, SSD, are equivalents to the angles which any plane parallel to the two normals makes with the two normal-chordal planes.

And the spherical angles 8 DD, S,D/D, are equivalents to the angles which any plane parallel to the two lines S.Z.., S..4,, makes with the normal-chordal planes.

The interpretation of the other points, lines, angles, and planes of the figure can present no difficulty, and no further elucidation is necessary here; but in order to avoid miscon- ceptions, it should be remembered that all through this paper (when two stations only are considered) we will consider the latitude of the station S, greater or not less than the latitude of the station S,.,—as indicated in the figure.

NOTATION.

, 4, denote the latitudes of the stations S,, 8,,, respectively.

ae a colatitudes, or the arcs PS,, PS

i, 1” _ arcs PD, PD,,.

eA.) azimuths or angles PS,D,, PSD.

Aes L,; angles PS|S,, PS_S,, of the triangle 8,PS,,. Pe Dy 5, oe PDS) Sb Dy Se oe - arcs §,D,, 8,,D,.

4 99

TP)

+ On Practical Geodesy. a,, a, denote the angles of depression of the chord, or arcs 8,E,,

; 8,4, D0), ¥, the small ares $,D,, 8,,D,,. DO ia. (angled SSD vans, D,

5 pe a anaes normals SZ, S. sooo, terminating in polar axis. Q); Q, 79 lines eZine Si Aids. Rt angles IPS, and supplement of IPS, 8, k ss lengths of geodesic are and chord respectively. v denotes the arc 8S,,, or the angle between the normals. > § circular measure of the geodesic are s. 0 4, arc PI, or angle between the chord and polar axis. A oF angle S,IS,, between the normal-chordal planes. a » length of the earth’s equatorial radius. b 3 » polar radius. é earth’s eccentricity.

1. Values of geodetic constants, in. accordance with the dimensions of the earth as finally adopted by the Ordnance Department of Great Britain and Ireland.

a = 20926348 feet log. a = 7°3206934433

b = 20855233 feet log. b = 7°3192150463 = *0823719976978 log. ¢ = 2-9157795987 = 0067851460047 log. & = 38315591974

(1—e) = -9932148539953 —log. (I—e’) = 19970432059

LRN Salli ett te (<—)= 1-0068314987210 log. (=a) = 0 0029567941 |

©) = 0068314987230 log. (2) = 38345159915

The geodetic tables above referred to give also the logs. to 8 places of decimals of the normals terminating in | the polar axis for all latitudes from the equator to the pole. The well-known formula by means of which any of these normals is expressed in terms of the latitude to which it pertains is—

a a J1—é sin? Z : 2. The following relations are evident from the figure— Cp ln, Cos.) Cp, = Ry cost, () S,p,=R, i—e’) sin 1, ; S,.P,=R, (l—e’) sin l, (2) "Z,=RB, é sin , ; C.2;,=3) é sin 1, (3) Q? —(C,p,)? +-(8,0, +0, foo) = R?, oR e” sin” i; +F (4) Q?,=(C.p,) + (8.2, +0.Z i R?, bow oR, e sin’ l, ie ( )

4 5

On Practical Geodesy. 5

in which F is the same function of the latitudes in the equation (4) and (5). S.p, 8..p,,=(R, sin 7, BR, sin J,,) . (1—e’) (6) 4, C,Z,,=(R, sin % R, sin l,,). ¢ (7) SP, aad Sy ee ALi as (1—é) : é 3. From the expressions for the magnitudes of Q,, Q,,, we have R? + Q? = 2°R? (1 e’ sin4l,) + F = 2a + F; B+ Q2 = 2B2(1 —esin”,) + F = 2a? + F And therefore it is obvious that we have the relation— R? + Q? = BR,’ + Q,? (9) Hence it follows that if N be the middle point of the segment Z,Z,, of the polar axis intercepted by the normals, “we oo NS, = NS,, (73) And fin this it is obvious that the stations S., S,., are in the surface of a sphere whose centre is N, and that we have

BQ, | WE Q, (11) OT O

(See formulz 81 -A and 81:B in the sequel.) 4. If in each of the triangles Z,Z,,8,, Z,Z,,8,,, we

1°) oo~o?

express the base Z,Z,, in terms of the other two sides and the included angle, it is evident from (9) that—

Ky 1) cos 0,1 bu,” Ge cos or Ord Dy SRCOS One. vine ke, , RCOa IO, a Ee a Gy ag Sa Q. wy Ate aN OL (13)

absolutely; but im all ordinary cases they are equals to at least 10 places of decimals in their logarithms.

5. It is evident that the plane through the middle point N, of the segment Z,Z,,, perpendicular to the geodesic chord SNe must bisect this chord or pass through its middle point M. And therefore, since the portions NZ,, NZ,,, of Z,Z,., which lie on opposite sides of this plane are equals, it follows that the planes through Z,, Z,,, perpendicular to the geodesic chord 8,8,,, cut it in points T,, T,,, equidistant from its middle point M. Hence—

sina, = cos TS A. 5,8.

oo)

sina, = cbs), 8,4 8, T.;

6 On Practical Geodesy:

js By (4)

And since we suppose /, greater than J,, we know that R, is