Mr. Boots on the Analysis of Discontinuous Functions. 133 
T 
Le 7 
*\ . z«\—=— ——. 
2cos (5) sin (05) sin(/z)’ 
a theorem which is known to be true of I in its ordinary definition, when / lies 
between O and 1, but which, according to the definition of F adopted in this in- 
vestigation, is thus seen to be true universally. 
oT (rl) = (6) 
2. The circumstance of an integral becoming infinite between the limits of 
integration, gives rise, as is well known, to an imaginary term in the function 
expressing its value. When this term is rejected according to Cauchy’s rule, a 
discontinuity of form is observed between the cases of the integral here consi- 
dered, and those in which the integral does not become infinite between the 
limits. Under these circumstances the former of these two cases may be re- 
garded as a limiting one, and of different origin from the other, and this is the 
true explanation of the discontinuity. Thus (/ positive), 
dx cos( lx) 7 =k BG 2COs (Ud) wee 7 as 4 
ia (2° (2+ he) =o‘ > SRL = py in). (7) 
But if, in the first theorem, we change / into hy —1, we have 
Ay dx cos(lz) _ 
77) Ss ——— ay 
array at Rare 5, in (Ch) —W7— 15, cos(th); 
the second member of which includes the rejected term. By an analysis founded 
on Fourier’s theorem we find, however, that the two integrals in (7) are respec- 
tively limits of the following, 
v—ke—k° 
(why k {a —hy-ke?y 
= dxe* cos(/x) 
\ (eh) 
k and k’ vanishing. 
All these circumstances require to be noticed when we compare a definite 
integral with its development. Thus the second theorem of (7) may be obtained 
by developing in descending powers of x, and integrating each term separately 
by (5); but it cannot be obtained from the ascending development. Although 
and \ dave ™ coslx 
2d 
the reason, in this case, is obvious, we are not prepared to lay down any g cweneral 
rule, but we conceive that such a rule must recognise the distinctions which we 
have endeavoured to elucidate. 
