[716 
716] 
AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 
43 
3 seen that 
that the sign of \/(X) is taken so that ^j^x) * S e 3 ua ^ a P os ifi ve multiple of i t 
and this being so the integral is taken from the inferior limit a to the superior 
limit x, which is real. 
^fu by an 
Take x a linear function of y, such that for 
x — a, b, c, d, 
y = 0, 1, ~, oc , respectively, 
so that, x increasing continuously from a to d, y will increase continuously from 0 to oo. 
We have 
J2 _b-a.d-c 
d — b . c — a’ 
eing as in 
V real and 
b — d x — a 
^ b — a x — d’ 
^ d — a x — b 
d b — a x — d,’ 
'2 Ku\ 
^1’ or ’ 
i 7 „ d — a x — c 
1 — №y — ,; 
c — a x — d 
and, thence, 
V(y. 1 — y. 1 — k 2 y) — d ~ a ./( d ~ b ). / (X ) 
J c—a\/\c—a/ (x — d) 2 
u, dn u in 
where ^ is taken to be positive, and the sign of \J(X) is fixed as above. Then 
for y between 0 and 1 or > , y. 1 — y. 1 — k 2 y will be positive, and *J(y.1—y.l— k 2 y) 
will also be positive; but y being between 1 and ^, y. 1 — y. 1 — №y will be negative* 
ritude, viz. 
considered 
i a and b 
ie positive 
ve assume 
and the sign of the radical is such that ^ ^ is a positive multiple of i. 
We have moreover 
7 d CL , 7 ... doc 
and therefore 
dy //7 7 , dx 
^y.\-y.l-¥y)~‘ J(d h - c 
where *d(d — b.c — a) is positive ; or, say, 
f 77—i—-^-5 \ = 0(d ~b . c - a) f • 
J 0 V(y.l-y.l-%) v JaO(X) 
6—2