39 



gent series. " Thus, &c.," as quoted above.* The analogy 

 thus apparently established is traceable to an oversight, of 

 very easy detection, in the preceding integrations ; which, in 

 the correct form, will stand as follow : 



> 



1 



x-^dx = + 00 — 





1— m 



m 



.'. adding, 

 Or thus, 



I 



It 1 



xr^dx =: + <x> 



1 m 



x-^dx = 2 00 . 



-VI in 



1 



-C0+- 



.'. \ x-^dx =1 + 00 ) — (— 'X) -\ — 



J-OT \ mJ \ in 



m 



= 2oo . 



m 



But errors of a much more important kind occur in all the 



applications of definite integrals to the summation of diverging 



series : a mode of summation first, I believe, adopted by 



Euler, and very generally employed by subsequent analysts. 



A single example of this method will be sufficient to shew the 



character of the errors adverted to ; which, though so glaring 



as almost to obtrude themselves upon the attention, have not 



hitherto, so far as I know, been noticed by any writer. Any 



one of the examples given by Euler (Institutiones Calc, Diff".), 



and afterwards by Lacroix (Traite du Calcul. &c., tome iii.), 



will answer the present purpose : I shall take that at page 



573 of the English edition of the smaller work of Lacroix, 



viz. 



s = \.t~ \.2t''+l.2.3f-8ic. (6) 



which, Sir John Herschel remarks, is such that " however 



* De Morgan's Differential and Integral Calculus, p. 571. 



