159] 
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159. 
ON SOME INTEGRAL TRANSFORMATIONS. 
[From the Quarterly Mathematical Journal, vol. I. (1857), pp. 4—6.] 
Suppose that x, a, b, c and x', a', b', c' have the same anharmonic ratios, or what 
is the same thing, let these quantities satisfy the equation 
1 , 
1 , 
1 , 
1 
= 0; 
X , 
a , 
b , 
c 
x' , 
a' , 
b' 
c' 
xx', 
aa\ 
by, 
cc' 
this equation may be represented under a variety of different forms, which are obtained 
without difficulty; thus, if for shortness 
K = a (b' — c') (x' — a') + b (c — a') (x' — b')+c (a' — b') (x' — c'), 
then 
Mx = — \bc (b’ — c ) (x' — a') + ca (c' — a,') (x' — b') + ab (a' — b') (x — c')}, 
K (x — a)— (c - a) (a —b) (b' — c') {x' — a'), 
K(x — b)— (a — b) (b — c ) (c' — a') (x' — b'), 
K(x — c)— (b — c) (c — a) {a! — b') (x' — b'). 
Consider x, x' as variables; then 
K 2 dx = (b — c) (c — a) (a — b) (b' — c') (c' — a') {a' — b') dx'; 
let, d, d! be any corresponding values of x, x'; then 
1 , 
1 , 
1 , 
1 
= 0 
a , 
b , 
c , 
d 
a' , 
y, 
y 
d' 
aa', 
bb', 
cc', 
dd' 
C. III. 
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