[ 93 ] 



XVIII. Letter from Mr. Ivory to the Editors of the Philoso- 

 phical Magazine and Annals of Philosophy. 



Gentlemen, 



T SENT you a short paper a few days ago, and since that 

 ■*- time I have looked into your last publication, which con- 

 tains M. Poisson's remarks. I shall answer all his remarks 

 when 1 find leisure and inclination ; but, in order to lessen 

 the work cut out for me, it may be proper to add here, as a 

 supplement to my letter to Professor Airy, what I have further 

 to say of Laplace's method of investigation. 



In M. Poisson's demonstration the arbitrary function y' is, 

 in reality, treated as a constant quantity. Assuming that the 

 said function is always finite within the limits of the integral, 

 he makes it constant while 6' and \[/' vary a little from their 

 initial values, after which all the increments of the integral are 

 regarded as zeros ; so that, in fact, the demonstration is the 

 same whether y' be constant or variable. 



Now the proposition to be proved is not true unless J ~J_^ 



be a finite quantity in the whole extent of the integral. But 

 it may happen in some cases, that this quotient may be in- 

 finite even in the nascent state of the arcs 6' and \J/'; which 

 would be contrary to the procedure of M. Poisson. It is the 

 finite value of the quotient mentioned within the limits consi- 

 dered, and not any limited portion of the integral, that makes 

 the demonstration conclusive. 



What is said at p. 14 of this Journal for June, is rather be- 

 side the purpose, and seems contrary to the meaning of La- 

 place. I understand that the quotient alluded to, Mec. Celeste, 



livr. 11'"% No. 2, p. 26, is -'f^' and not -^^, the value of 



a mf- being extremely near 1. This latter view of the case 

 does not appear to answer the intention of the author ; for the 

 quotient would be evanescent when y' — y = 0, independently 

 of the nature of the functions. In my view, and especially if 

 the (|uolient is finite in the whole extent of the integral, the 

 demonstration of Laplace is exact, although very limTted. 



I have always maintained that the functions to which this 

 analytical mediod can l)e applied, are determinctl by the me- 

 thod itself. I Inive ascertained the nature of these functions 

 HI two (liflerent ways in my two letters to Professor Airy. 

 The fundiUMental jjroposilioii, or e<ju;ition, is true only when 



