Mr. Ivory's Remarks on M. Poisson's Memoir. 327 



of its improvements, and which ought never to be admitted 

 without scrupulous examination. In reality, the pi'ocedure of 

 M. Poisson hides, from the attention of his readers, the true 

 principles of the case. The numerator and the denominator 

 of the expression vanish together; and the value of the fluent 

 will depend entirely on the limit of the ratio of the two quan- 

 tities as the}^ both approach zero. According to that value, 

 the fluent may be evanescent, or it may be finite, or infinitely 

 great. 



It is remarkable that the analytical process employed by 

 M. Poisson, if he had pursued it accurately, would have led 

 him to a right result. Put fl' = fl + //, \|/' = vj/ + /t ; then ac- 

 cording to the operations in p. 331, the tei'm of which the 

 value is sought, will take this form, viz. 



^ /^ S {y ~~ y) ^i** 6dhdlc 



But as y varies with h and k, we must not make y' — y, or the 

 ? of M. Poisson, constant. We have 



putting A and B for the differential coefficients. By substitu- 

 tion, our expression will become, 



1 ^ g Asm eh dhdk 



— r- 



+ A^ + ^^sin2^)2 

 g'Q%\r\.6dhhdk 



Here the two parts are similar, and the integrations are readily 

 performed by the procedure of M. Poisson : the result is this, 



_I_ j Alog.^^^^li^ + ^, 1 ^^+i!f!^^ 



Although g is a vanishing factor, we must not immediately 

 infer that the whole of this expression is always evanescent. 

 It is necessary to take into account the ultimate values of 



— and --, which again depend upon the limit of^-^. If we 



suppose that y — 3/ is ultimately divisible hyj'% it is manifest 

 that the expression is evanescent, which proves M. Poisson's 

 proposition for such functions. This is the only case compre- 

 hended in Laplace's demonstration, Mec. Celeste, liv. xi. pp. 25, 

 26. Again, if we suppose that y^ — y is ultimately divisible 

 by^i the quantity multiplied by g^ will be finite; the whole 

 expression will therefore be equal to zero; and this proves 

 the proposition for all rational functions of cos fl, sin 6 cos (p, 



sin 



