422 
on MR Wilkinson’s rectangular transformation. 
[783 
and consequently 
(p + v) (;yz' — y'z) 2 + (q + v) (zx — z xf + (r 4- v) (xy' — x'y) 2 = 0. 
It is to be shown that this equation is implied in the remaining first, second, and 
third equations; for, this being so, (x, y, z), {x, y', z') satisfy only these equations; 
or (x, y, z) are any values whatever satisfying the first equation. The other two 
equations then determine (x, y\ z'), and, these being known, (x", y", z") are then 
determined as above. 
In fact, attending to the sixth equation, the equation just obtained may be 
written in the form 
(p 4- v) [(y 2 4- z 2 ) (y 2 + z' 2 ) — x 2 x' 2 ~\ + (q 4- v) [(z 2 + or) (z' 2 + x 2 ) — y 2 y' 2 ] 
+ (r + V) [(x 2 + y 2 ) (x 2 + y 2 ) — z 2 z' 2 ] — 0, 
or, what is the same thing, in the form 
— (x 2 4-ÿ 2 + z 2 ) [{p 4- /¿) x' 2 4- (g 4- At) y 2 + (r + fi) z' 2 ] 
- (x 2 4- y 2 4- z' 2 ) [(jp 4- X) x 2 4- (q 4- X) y 2 4- (r 4- X) z 2 ] = 0 ; 
for, comparing in the two forms, first the coefficients of x 2 x 2 , these are 
(q 4- v) 4- (r 4- v) — (p 4- v) and - 2p 4- X 4- At, 
which are equal in virtue of p + q + r + \ + jLL + v = 0) and comparing next the 
coefficients of y 2 z' 2 , these are 
p + v and - (r +p) — (q + X), 
which are equal in virtue of the same relation : and, similarly, the coefficients of the 
other terms y 2 y' 2 , &c., are equal in the two equations respectively. 
Take now three arguments a 0 , b 0 , c 0 , connected by the relation a 0 4-6 0 + c 0 = O, 
and write a, a, A for the sn, cn, and dn of a 0 ; and similarly b, b, B and c, c, C 
for those of b 0 and c 0 respectively : then we may write 
for, starting from the first set of values, we have the second set if only 
We thence obtain