Ml*. Ivory's Remarks on M. Poisson's Memoir. 325 



and acquirements of the author of tlie Memoir are well known : 

 he has made this branch of the mechanical philosophy more 

 particularly his study, and has applied the peculiar kind of 

 analysis employed in it to different problems ; so that among 

 existing mathematicians an abler vindicator could not have 

 been found. Every subject that passes through such hands 

 must acquire valuable improvements ; and if, on the present 

 occasion, M. Poisson has not succeeded in removing every 

 difficulty, this must be ascribed to the doctrine he defends, 

 which can never be entirely freed from inconsistency, nor per- 

 fectly reconciled to clearness and accuracy of demonstration. 



The author begins his Memoir with stating anew the fun- 

 damental principle of the analytical theory : he then repeats 

 the demonstration of it he had given on a former occasion, 

 and endeavours to defend it against an objection advanced 

 in the Phil. Mag. for January 1826, p. 37. Tt is chiefly on 

 this part of M. Poisson's paper, extending through about four 

 pages, and another short passage, that I intend to offer some 

 brief remarks. Supposing that the reader has the Memoir 

 alluded to before him, I shall, for the sake of abridging, write 

 ij and y for ^ (9, vj/) and <p (9', 4^'), and / for \/ 1 — 2 up + u": 

 I shall also write ds for the differential of the surface of the 

 sphere; it is equal to sin ffd6'di>', when its position is deter- 

 mined by the arcs fl' and 4*' ; and it may be similarly expressed 

 by any other two independent arcs that fix its place, if any 

 transformation should make this convenient. For the sake of 

 simplicity, I shall further suppose that « never exceeds 1, al- 

 though it is always near it, and approaches it as a limit. The 

 formula (3), p. 330, will then be thus written, 



>(1 - a.^')y'ds 



4^y 



Now the fluent being extended to the whole surface of the 

 sphere; or, which is the same thing, the integration being 

 eifected separately for the two variables fl' and ^', from S' =. 0, 

 •y = to fl' = TT, \J;' = 2 TT ; it is proposed to prove that X = j/, 

 in the particular case when « = 1. 



The distinguishing features of the formula are these : the 

 numerator is always inconsiderable, because 1 — a" is small; 

 ami, taking 9 and \I/ lor the initial values of 9' and \J;', the de- 

 nominator increases raj)idly from the least value (1 — a)^, so 

 as to become incomparably greater than the numerator when 

 tlie two variable arcs have acquired very small increments. 

 On these grounds M. Poisson thinks himself entitled to inte- 

 grate on the assumption that y' does not vary from the initial 

 value J/: then, \ — JL /• ( ' -«■')<{ •< 



4t./ /3 



The 



