Jan. 1 8, 1877] 



NATURE 



253 



by those functions, and we have a long disquisition on 

 the geometrical representation of the elliptic integral of the 

 first kind by an algebraic curve ; while there is no mention 

 of the late John Riddle's discovery, that the arcs of the 

 curv^es by which circles on the sphere are represented in 

 Mercator's projection are directly given by, and abso- 

 lutely co-extensive with, the elliptic integrals of the first 

 kind, the amplitude being simply the longitude on the 

 sphere. We think this quite as simple and as important 

 as the discussion of the lemniscata. If we are to go into 

 geometry at all, it might be as well also to make some 

 allusion to Dr. Booth's discussion of the spherical conies, 

 and to Mr. Roberts's integration of the Cartesians. 



Then, again, we have an account of Jacobi's geo- 

 metrical theorem in its original form, depending upon a 

 family of circles having the same radical axis, while the 

 corresponding theorem, depending upon circles having 

 two inverse points in common, given by Chasles (see his 

 "Gdometrie Supdrieure," cap. xxxi., p. 533), which much 

 more directly represents both amplitudes and moduli, is 

 not mentioned explicitly, although it is involved in the 

 geometrical exposition given of Landen's theorem. 



We have also been unable to find any account of 

 Jacobi's reduction of the integral of the third kind to the 

 form — 



fd^. E<p : Ac/) 



The transformations of ihe functions are worked out 

 with great completeness, the results being tabulated in 

 some rather formidable-looking, but really very con- 

 venient, schedules. This part of the work is carried 

 almost to an extreme. 



On the whole the book is one of the most important 

 contributions to mathematical literature which has ap- 

 peared for a long time. It is well done, and covers 

 ground that was previously but ill occupied. It is clearly 

 printed, and the fact that the proof-sheets have been 

 revised by Mr. J. W. L. Glaisher is a guarantee for the 

 correctness of detail. C. W. MerrifielD 



Ol/R BOOK SHELF 



Instruction in PJiotography. By Capt. Abney, R.E., 

 F.R.S., &c., Instructor in Chemistry and Photography 

 at the School of Military Engineering, Chatham. Third 

 Edition. (London : Piper and Carter, Gough Square, 

 Fleet Street, E.C, 1876.) 



We are very glad to find that Capt. Abney did not carry 

 out the intention which he mentions in the preface of not 

 producing another edition of his well-known ** Instruction 

 in Photography." That the little volume is widely known 

 and appreciated is shown by the fact of its having reached 

 a third edition, and we can only say that it well deserves 

 its success. A photographer of the author's well-known 

 skill and repute could not fail to be able to instruct others 

 in his art, but when in addition he has gained large ex- 

 perience by continued practical teaching in such a school 

 as that at Chatham his lessons become additionally 

 valuable. 



Capt. Abney does not enter much into theory, though 

 he gives very good and simple accounts, illustrated by 

 chemical equations, of the principal changes occurring 

 during the processes described. We observe that he 

 announces the forthcoming publication of a " Photo- 

 graphy" among Messrs. Longmans' Text-books of Science, 

 in which he proposes to deal more fully with the theoreti- 

 cal part of the subject. We shall look forward to this 



with considerable interest ; meanwhile, for practical in- 

 struction in the art this little book distances all com- 

 petitors. R. J. F. 



LETTERS TO THE EDITOR 



\The Editor does not hold himself responsibte for opinions expressed 

 by his correspofidents. Nather can he undertake to return, 

 or to correspond with the writers of, rejected manuscripts. 

 No notice is taken of anonymous communications. '\ 



Just Intonation 



The errors and oversights — in my paper in Nature, (vol.xv. 

 P- '59^ — with which Mr. Chappell charges me, are imaginary. 

 To make the matter clearer, the vibration numbers of a diatonic 



scale started from ^ as a tonic are — 



2 



5 45 , 



3 27 15 2 9 

 2 16 8 4 



16' 



In order to keep to the same part of the keyboard, let the last 

 five notes be depressed one octave, and we get this series : — 



3 27 15 



I 9 5 45 



' 8' ? 32' 



2' 16' 8' 



2, 



where 3. is the tonic, and i or 2 the subdominant. A similar 

 2 



explanation applies to the scale starting from - as tonic. With 



respect to the " comma of Pythagoras," I am not aware of any 

 "generally adopted miscalculation." Who was the real dis- 

 coverer of that interval is a matter of no consequence ; but in a 

 system such as the Pythagorean, which was tuned by true fifths, 

 it would have been not only a very natural but an essential in- 

 quiry, " What definite number of fifths corresponds with another 

 number of octaves?" This, without at all necessitating the 

 supposition of the existence among the Greeks of instruments of 

 an immense range in octaves, would be but an easy arithmetical 

 calculation resulting necessarily in the conclusion that there was 

 no exact coincidence between fifths and octaves, but that twelve 

 fifths differed by a very small quantity from seven octaves. This 

 small difference is therefore very aptly termed the Pythagorean 

 comma. Now in the equal temperament system twelve fifths 

 just coincide with seven octaves, so that the despised comma of 

 Pythagoras is really a measure of the error of the equal tempera- 

 ment fifth. In fact, putting P for the comma in question — 



(i)' 



so that if 2 N be the vibration numbsr of the lower tone of an 



_ 1 

 E T fifth, that of the upper tone will be 3 N P '^'^, which is in 



error by 3N(i -P ') vibrations, giving rise to a number of 



Beats = 6N(i - P"'''') 

 per second in the E T fifth. Not, however, that it is necessary 

 to allude to Pythagoras or seven octaves to get those beats. 



The grounds upon which Mr. Chappell declines to accept 24, 

 27, 30, 32, 36, 40, 45, 48, as representing the vibration numbers 

 of the diatonic scale are not very clear, certainly ; and repudiat- 

 ing these, he can of course have no sympathy with Colin Brown's 

 keyboard. 



To revert shortly to this, the subject of my previous commu- 

 nication, —in his new and very interesting work on " Tempera- 

 ment," Mr. Bosanquet has given a description of Colin Brown's 

 keyboard, but in so peculiar a manner that it is really difficult to 

 recognise the instrument at all, and neither its elegance nor sim- 

 plicity are brought out as I think they should be. 



A. R. Clarke 



Ordnance Survey, Southampton, January 9 



South Polar Depression of the Barometer 



I THINK it probable that on this subject Mr. Murphy's views 

 and my own might have appeared more in harmony if we ha I 

 neither of us expressed them with so much brevity. In my 

 letter on Ocean Currents in Nature, vol. xv. p. 157, I was 

 incidentally lad to speak of the barometric depression round the 

 South as greater than that round the North Pole. In speaking 



