424 
on MR Wilkinson’s rectangular transformation. 
[783 
which is easily shown to be 
so that the values of x, y, z are 
= k 2 k' 2 bc (a 2 — f 2 ), 
= {Vk' 2 Aa, Vk 2 Ahc . f, V— aBG. F] ■— VIc 2 k' 2 bc (a 2 — f 2 ), 
and, similarly taking the arguments g 0 , h 0 , and denoting their elliptic functions by 
g, g, G, h, h, H, we have for a system of arbitrary lines in the three cones 
respectively, the values 
x, y, z 
x', y, z 
VF 2 ffa 
Vk 2 À be .f 
V-aBC.F 
VFiëb 
\Zk 2 Bcsi . g 
V- b G A . G 
V k' 2 Gc 
V k 2 Gab . h 
V— c AB. H 
-r- Gk 2 k' 2 bc (a? — / 2 ) 
-T- Vk 2 k’ 2 ca (b 2 — g 2 ) 
4- \fk‘ 2 k t ' 2 ab (c 2 — A 2 ), 
these values being such that x 2 + y 2 + z 2 , x' 2 + y' 2 + z" 2 , x" 2 + y" 2 + z" 2 are each = 1. The 
radicals in the first line would be more correctly written, and may be understood as 
meaning k' k \/b \/c, i \/a \JB \]G, and similarly as regards the second and 
third lines respectively. 
Taking now the arbitrary lines at right angles to each other, the condition for 
the second and third lines is 
1 + & 2 agh — k' 2 A GH = 0, 
which is satisfied if a 0 = g {) — h 0 ; similarly the condition for the third and first lines 
is satisfied if b 0 = h 0 — f 0 ; and we then have a 0 + b 0 = g 0 — f 0 ; that is, — c 0 = g 0 —f 0 or 
c 0 =fo~go, which is the condition for the first and second lines; hence the arguments 
ao, b 0 , c 0 , f 0 , g 0 , h 0 being such that 
• K - go + «o = o, 
— h 0 . + > /o+A 0 = 0, 
9o -fo • + C 0 = 0, 
n 0 b 0 Co . — o, 
or, what is the same thing, a 0 , b 0) c 0 , f 0 , g 0 , K being the differences of any four 
arguments a, ¡3, y, 8, the foregoing values of (x, y, z), {x\ y, z'), (x", y", z") will 
satisfy the equations 
x 2 +y 2 + z 2 = 1, 
x' 2 + y’ 2 +/ 2 = 1, 
x" 2 + y" 2 + z" 2 = 1, 
x’x"+ y'y"+ z'z" = 0, 
x''x + y"y + z"z = 0, 
x x' Ay ÿ + z z' =0, 
for the transformation of a set of rectangular axes. These are, in fact, Mr Wilkinson’s 
expressions, the a 0 , b 0 , c 0 , f 0 , go, K being his t — p, p — q, q — t, t, p, q respectively.