423 
783] on MR Wilkinson’s rectangular transformation. 
and, in order to the identity of the two values of 6, we must have 
that is, 
(abc - b 2 ) (ABC - A 2 ) - (abc - a 2 ) (ABC - B 2 ) = 0, 
or, reducing, 
(a 2 — b 2 ) ABC — (A 2 — B 2 ) abc + J. 2 b 2 — B-a 2 = 0. 
But 
a 2 — b 2 = — (a 2 — b 2 ), A 2 — B 2 = — k 2 (a 2 — b 2 ), A 2 b 2 — B 2 a 2 = k' 2 (a 2 — b 2 ); 
hence the whole equation divides by a 2 — b 2 , and, omitting this factor, it becomes 
— ABC + /c 2 abc + k' 2 = 0, 
which is a known relation between the elliptic functions of the arguments a 0 , b 0 , c 0 
connected by the equation a 0 + b 0 + c 0 = 0. Similarly, for </>, we have 
v- = ° 
_A 
AB BC ’ 
ab be 
and, comparing the two values of <£, we have the same identical relation. 
It thus appears that the three cones 
X!+ c- a F!+ (3^= 0 ’ 
Z2 + ^b y, + Z8*-°- 
(the coefficients whereof depend on the elliptic functions sn, cn, and dn, of the 
arguments a 0) b 0 , c 0 connected by the equation a 0 + i 0 + c 0 =0) contain a singly infinite 
system of rectangular axes. 
Considering an argument /„, and denoting its sn, cn, dn by /, f, F respectively, 
we have, for an arbitrary line on the first cone, the values 
x, y, z — M^k' 2 Aa, M\f^ 2 JLbc. f, M V— aBC. F. 
In fact, substituting in the equation of the cone, we obtain the identity 
k' 2 + k 2 i 2 -F 2 = 0; 
and if we determine M by the condition that x 2 + y 2 + z 2 shall be = 1, then we have 
1 = M 2 {k' 2 Aa, + k 2 Abci 2 - aBCF% 
where the coefficient of M 2 is 
= k’ 2 Aa, + A^vlbc (1 — f 2 ) — aBC(l — k 2 f 2 ),