Mr. Boo.e on the Analysis of Discontinuous Functions. 127 
according as x lies between, upon, or without the limits a and a+ Aa. It 
we wish that f(a’) should vanish for values of # external to p and gy, we shall 
have 
=} tan (*) a tan f(a) = f(x) or aS or 0, (6) 
according as x lies within, upon, or without the limits p and 4. 
As regards the mere expression of discontinuity, this formula is on a par with 
those which will follow, but it is wanting in certain other advantages which they 
possess. 
3. When Aa becomes infinitesimal it may be replaced by da, and the symbol 
A by d, whence, by (4) and its consequences, 
a—x re 
=7 or = or 0, (7) 
J tan- = 
a tan ke 3 
according as v hes between, upon, or without the limits a anda + da. Effecting 
the operation in the first member, we have, under the same conditions, 
kda 7 5 
SFP = OF Ue 
ke +-(a—wx)? 2 
and since, under the two first conditions, the values of a and @ are indefinitely near 
to each other, 
1 kdaf(a) 
Jie aae g 
ae (aa) = f(Z) or-—— or 0. (8) 
= 
AC 
Extend this by integration from p to q, then observing that each half value,—>— oa == 
occurs in two contiguous elements, except the first of them and the last, we have 
kedafla) 7 », fla) 
ee (9) 
according as « lies within, upon, or without the limits p and q. 
If p and g are respectively — » and o, then, for all real and finite values 
of x, 
1c _ kdaf(a 
eS = sla). (10) 
