620
ON THE LINEAR TRANSFORMATION OF THE INTEGRAL
[629
The relation between u, v! may be written in the forms
U — O,' -pU — Cl ll' — b' _ n U — b U — c' _ Tf U - c
u' — d! u — d’ u' — d' * u — d’ it — d! u — d’
and then, writing for u, u' their corresponding values, we find
P — k’h _ c'g p. __ c'f _ a'h „ _ a'g _ b'f
bh /_ c^'’ ^ ~ ag' — bf' ’
giving
PPN* = i' 2 QR, g 2 QJST 2 = g' 2 RP, h 2 RN 2 = h' 2 PQ, ^PQR = ^,N\
i g n
Differentiating any one of the equations in (a, 10), for instance the first of them,
we find
i'du' _ fPda
(u' -dj~ (u-d) 2 ’
then, forming the equation
Ve. v! — a’. v! — b'. v! — c'.u' — d'_ V PQR *Je.n — a. u — b.u —c.u — d
(u' — d') 2 ~ (u — d) 2
and attending to the relation i 2 PN 2 — i' 2 QR, we obtain
Ndu' _ d a
± ^/^F z= 7u ,
which is the differential relation between u, u\
We have in connection with A, B, G, D the point 0, and in connection with
A', B', G', D' the point O'. As U describes the right line AB, U' describes the
arc not containing 0' of the circle A'B'O'; for observe that 0' corresponds in the second
figure to the point at infinity on the line AB, viz. as U passes from A to B, not
passing through the point at infinity, U' must pass from A' to B', not passing through
the point O', that is, it must describe, not the arc A'O'B', but the remaining arc
2tt — A'O'B', say this is the arc A'B'. The integral in regard to u' is thus not the
rectilinear integral {A'B'), but the integral along the just-mentioned circular arc, say
this is denoted by {A'B')] and we thus have
{AB) = ± N{AB').
But we have {A'B') = or not = {A'B'), according as the chord A'B' and the arc
A'B' do not include between them either of the points G', D', or include between
them one or both of these points; and in the same cases respectively
{AB) = or not = ±N{A'B').
Of course we may in any way interchange the letters, and write under the like
circumstances
{AG) = or not =±N{A'G'), &c.