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 : 
\ a : 
Ps “otiy DB nes 
] 
Oe ae ya 
) adz = +0 > 
“, adding, 
a“*dz = 20 — ed 
—m m 
Or thus, 
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 Cale. 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 = 1.t—1.2241.2.3°—&e. (6) 
which, Sir John Herschel remarks, is such that ‘ however 
* De Morgan’s Differential and Integral Calculus, p. 571. 
