Letter frrnn Mr, Ivory to Professor Airy. 89 



when a= 1, I say that X =■ y, whenever 2/ — y is divisible by 

 \/ I — p, or whenever the quotient of the same division has 

 always a finite value; but, if this condition do not hold, then 

 X is indeterminate. 



It is to be observed that if we make y' equal to unit, or to a 

 constant quantity in the foregoing formula, X will be equal to 

 unit, or to the same constant quantity. This arises from the 

 nature of the fluent, which can be integrated without difficulty 

 by the ordinary rules. M. Poisson employs a particular mode 

 of integration, which, as it leads to the same result, makes no 

 diflerence. Write y + [y' — y) for y' ; then, according to what 

 is just observed, we shall get, 



We have next to consider tiie term under the sign of integra- 

 tion, which I write in this manner, 



J 



Beginning on the left, the first factor is always finite in the 

 hypothesis laid down : the last factor too is never infinite ; it 

 is a constant differential multiplied by a finite quantity : the 

 second factor never exceeds 1, and the third never exceeds 

 2. But although the third factor is exactly equal to 2, when 

 a = 1 and p = 1, yet, when «= 1, it is evanescent for 

 every value of ^j less than 1. It is evident, therefore, that 

 the integral, as it receives no accumulation by the increase 

 of the arcs 9' and 4/', is equal to zero. Wherefore we have 

 X=j/. 



The demonstration now given rests entirely on the sup- 

 position that y— J/ is divisible by V I — p. For instance, 

 ify' — y were divisible by (1 — pf, e being less than h, the 

 first factor on the left would be infinitely great, anil nothing 

 could be affirmed concerning the value of the integral, or of 

 X. It is therefore essential in this analytical theory thaty' — y 



be divisible by \/ 1 — p. 



I must here remark, that the strictures I made in this Jour- 

 nal for May last, pp. 32fi, 327, 328, on M. Poisson's demon- 

 stration, are perfectly correct. These strictures arc founded 

 on the condition I there deduced by following his analysis; 

 and that condition I have here made the prmciple of the fore- 

 going demonstration. On examining what I had written, I 

 found it was liable to be misunderstood ; and, despairing to 



Ni-jo Series. Vol. 2. No. 8. An<y. 1827- N i,iiake 



