200 Mr. A. Cayley on an Analytical Theorem connected with 



that the investigation jorores too much, viz. it appears to prove that 

 y is actually equal to zero. There are undoubtedly functions such 



as the function e"^ (noticed by Cauchy and SirW. R.Hamilton), 

 which in a sense have the property in question, viz. that if we 

 attempt to develope them in positive integer powers of x, the 

 coefficients are found to be all of them zero ; and it would appear 

 that y is, in regard to \ — fi, a function of this nature. But it 



cannot be asserted simpliciter that e"*^ and its differential coeffi- 

 cients do in fact vanish for a?=0; they only vanish iov x = 

 considered as the limit of an indefinitely small i-eal positive or 

 negative quantity. (This is quite consistent wath a remarkable 



1^ 



theorem of Cauchy's, by which it appears a priori that e .«' can- 

 not be expanded in positive integer powers of x, because it is 



discontinuous for the modulus zero.) And if, instead of a direct 



_\_ 

 application of Maclaurin's theorem, we first expand e'x'', say in 

 positive powers of 1 —x, and then develope the several terms in 

 powers of x, wc obtain for the coefficient of x^, or any other 

 power of X, an infinite series, which I apprehend is not conver- 

 gent, and which can only be equal to zero in the same conven- 

 tional sense in which e~'^ is equal to zero for x = 0. This ap- 

 pears to be something very different from finding for the coeffi- 

 cient of a?°, or of any other power of x, an expression composed 

 of a finite number of finite terms the sum whereof is identically 

 equal to zero. 



Plana has given for the calculation of y when /u- is nearly equal 

 to 1, an expression (equation (127)) which is deduced from the 

 same development of TJp which is here made use of; but it 

 appears to me that this expression is, for the following reason, 

 open to objection. The expression referred to contains explicitly 

 positive and integer powers of fju, and also powers of the radical 

 ^^2^2(1— /i)(l +b) : it would be, for anything that appears 

 to the contrary, allowable to develope as well the positive and 

 integer powers of /i. as also the powers of the radical in question, 

 in a series of positive and integer powers of (1— /u.) ; but if this 

 were done, we should obtain as a mere transformation of Plana's 

 expression (127), an expression for y developed in a series of 

 positive integer powers of (1— /a); and for consistency with the 

 before-mentioned result, the coefficients of the different powers 

 of 1 — /-t must be each equal to zero. But if this be so, it does 

 not appear how the original exjn-ession (127) can be anything 

 else than zero. The difficulty is, I think, a real one ; and I do 

 not see how it is to be got over: it seems to render necessary a 

 more careful study of the effect of the multiplication of the sue- 



