1 36 Notices respecting New Books. 



quality, I do not approve of the system; and moreover, I think 

 that a chronometer which makes a great extreme variation in 

 any month's rate, would not be so much affected by it. I do 

 not see the necessity of squaring the differences as proposed 

 by J. L. T., but I think it would be very laborious ; as for ex- 

 act calculations the mean rate must be given to the hundi'edth 

 part of a second. 



Having now suggested my system for deciding the quality 

 of a chronometer, I leave it to the investigation of those who 

 are willing and competent to decide upon its merits. Trusting 

 my observations may prove useful, I remain 



Your most obedient servant, 

 January 20, 1830. R. J. M. 



XX. Notices respecting Netv Books. 



On the Theory of the Small Vibratorij Motions of Elastic Fluids, by 

 J. Challis, M.A., Fellow of Trinity College, and of the Cam- 

 bridge Philosophical Society*. 



THE following is the introduction to Mr. ChalJis's memoir, from 

 which the reader will obtain a correct view of the importance 

 of the investigation which it contains. — " Any one that has given 

 much attention to the mathematical theory of sound, will be aware 

 that notwithstanding the labours of the most eminent mathematicians, 

 great obscurity is still attached to it. Much of this obscurity, I have 

 been led to think, is owing to the manner in which discontinuous 

 unctions have been introduced into the subject ; and as geometers 

 of late have been more engaged in the use of them than in scruti- 

 nizing the evidence on which they rest, I will endeavour to state, as 

 briefly as possible, the nature of this evidence. It depends, I believe, 

 almost entirely on the authority of Lagrange, and on his two dis- 

 sertations contained in the first and second volumes of the Miscel- 

 laneaTaurinensia. His first Researches, however admirable in other 

 respects, cannot be adduced in reference to the point before us, be- 

 cause that part of them which bears upon it, contains a step in the 

 proof which can by no means be admitted. In fact, it mainly depends 

 on the sum of the series cos Q + cos 2 + cos 30+ &c. ad infini- 

 tum, which he determines to be always equal to — ■§. And in truth, 

 if the exponential expressions be put for the cosines, and the series 

 be summed to infinity, this result is obtained. But the objection is, 

 that a mode of summing a converging series is applied to one whicli 

 is not convergent. The only legitimate method is to sum the series 

 to m terms, and to find what the sum becomes when m is infinite. La- 

 grange does this ; he finds the sum to be cos>»e-cos (m + l)0 _ , 



2(1— cosy) 

 and says, that the first term disappears when m is infinite, because 



* From the Trnnsactions of the Cambridge Philosophical Society. 



the 



