Mr. Booue on the Analysis of Discontinuous Functions. 129 
4. Resuming the formula 
] \" kdaf(a) 
bia) a)_,K+ (a—z)” 
and observing that 
BR heii te nome, PSO NEY 
k4(a—ry ~~ *¢k+ (a—z)/—=1 | k—(a—2)/—13 
— 3 {cE He a9 dy <5 emer ai dv} 
= ‘a « cos(av—.rv) dv, 
we have, on substitution, 
: Mere (es E : . 
ia) = 20 dadve cos(av—wv) f(a), (12) 
which may be written in the form 
Kays = RN dadv cos(av—.axv) f(a), (13) 
provided that the second member be regarded as an abbreviated form of the 
second member of the equation preceding ; according to which assumption 
{, dvg(v) is to be understood to mean the limit of (, ve" (0) for positive 
diminishing values of k. We may call this the extraordinary, or limiting mean- 
ing of the symbol §. . 
But is the formula (13) true, when to the second sign of integration we 
attach its ordinary meaning? If it is true, it must be so either in virtue of the 
principle of continuity assumed @ priori, or because the proposition which is 
affirmed of the integrals is true universally of their elements. The former 
assumption we dismiss; the latter requires us to consider whether the limit of 
« cos(av — rv) is cos(av — av), for all values of v from 0 to » included. It 
is so for all finite values of v; but when v is infinite, the limit of the first ex- 
pression is 0, since the value of e— cos(av—kv), for every positive value of k, is 
nothing. But the second expression becomes cos, respecting which, if we 
were entitled to make any assertion, it would be that it is indefinite. When a 
definite value is assigned to cos ®, it is through that assumption of continuity of 
which we have already spoken, as involving an unauthorized induction. We 
VOL. XXI. s 
