783] 
421 
783. 
ON MR WILKINSON’S RECTANGULAR TRANSFORMATION. 
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883), 
pp. 222—229. Read May 10, 1883.] 
where 
Considering the three cones, 
(p + X) X 2 + (q + X) F 2 + (r + X) Z 2 = 0, 
(p + (i) X 2 + (q + /a) Y 2 + (r + fi) Z 2 = 0, 
(p + v) X 2 + (q + v) Y 2 + (r + v) Z 2 = 0, 
p + q + r +X + fi + v= 0, 
it is easy to see that these contain a singly infinite system of rectangular axes, 
viz. we have in each cone one axis of a rectangular system, and for one of the 
cones the axis may be any line at pleasure of the cone. In fact, taking for 
the three axes (x, y, z), (x, y, z), («", y", z") respectively, that is, for the first 
axis X : F : Z — x : y : z, and so for each of the other two axes, then (x, y, z) 
being an arbitrary line on the first cone, we can find (x\ y\ z) and ix', y", z") such 
that 
(p + X)x 2 + (q + X) y 2 + (r + X) z 2 = 0, 
(p + fi) x 2 +(q + y) y' 2 + (r + fi) z' 2 = 0, 
(p + v) x” 2 + (q + v) y" 2 + (r + v) z" 2 = 0, 
x' x" + y y" + z' z" = 0, 
x"x + y"y + z" z =0, 
x x' +y y' +z z' = 0. 
For, eliminating (x", y", z") from the third, fourth, and fifth equations, we have, 
first, 
x" : y" : z" — yz' — y z : zx — z'x : xy' — x'y,