AS COMMONLY INVESTIGATED. 225 



function is alone sufficient to overthrow the customary demonstrations of Tay- 

 lor's Theorem, and rests the necessity for his own method of proceeding upon 

 the evidence it affords. It is yet manifest that every transient function must 

 possess the same "apparent" anomaly of making the function itself, and all its 

 differential coefficients, vanish for a given value of x; and, indeed, the whole 

 doctrine of what Cauchy has termed singular integrals rests upon the same 

 foundation. The anomaly observed by the first of the authors we have cited 

 consists in the fact that, whilst the function vanishes for x = 0, it cannot be ex- 

 panded in a series proceeding by the powers of x. 



Mr. Peacock, having adopted the maxim that all developments may be re- 

 garded as universally true, is led to hope, in noticing this function, " that more 

 enlarged views of the analytical relations of zero and infinity, and of the inter- 

 pretation of the circumstances of their recurrence, as well as of the principles 

 and applications of Taylor's series, may enable us to explain these and other 

 anomalies."* Without doubt, they are very easily explained in the present 

 instance ; and it may be matter of surprise that neither Sir W. Hamilton nor 

 Mr. Peacock have noticed the explanation. I have not, indeed, at present, the 

 Irish Transactions by me, and am, therefore, obliged to restrict myself to the 

 analysis and remarks contained in the work here quoted, but which, I presume, 

 are drawn from the original memoir.f The method used is to put the theorem 



e~^ = Ax'' + Bx'^-l-Cx>'-f- &c. 



under the form 



_ 1 



■ X *. e~^ = A + B x^-* -f C x>-* -f- &c. (a) 



When, assuming y = - , expanding 



A. 





4 — a. 



{x .e x.|-^ = y-^e 

 under the form 



J ~ " + y ' ~ ^ + FTF— + <^^" 

 and observing that when x ;= or y = oo, this series is infinite, it is concluded 

 that the left hand member of (a) must be zero, and consequently A = 0; a re- 



(b) 



* Third Report, p. 846. 



t I have since found this surmise correct, having received the Transactions in the interval that 

 elapsed in going to the press. 

 VII. — 3 G 



