783] on mr Wilkinson’s rectangular transformation. 427 
where the quantities (x, y, z), as belonging to a line on the first cone, satisfy the 
condition px 2 + qy 2 + rz 2 = 0. The equation may be written 
(a, b, c,f g, h\yZ— zY, zX-xZ, xY-yX) 2 = 0, 
where 
a, b, c, f g, h = q'z 2 + r'y 2 , r'x 2 + p'z 2 , p'y 2 + q'x 2 , — p'yz, — cqzx, — r'xy, 
and, as before, px 2 + qy 2 + rz 2 = 0; viz. this is the equation of the pair of planes (12) 
and (12'). 
The equation of the pair of tangent planes through the line (x, y, z) to the 
cone q"r"X 2 + r"p"Y 2 +p"q"Z 2 = 0 is 
(q"r"x 2 + r"p"y 2 + p"q"z 2 ) (q"r"X 2 + r"p"Y 2 + p"q'Z 2 ) - (q'Y'xX + r"p"yY + p"q"zZ) 2 = 0; 
viz. omitting a factor p"q"r", this equation is 
(p", q", r", 0, 0, 0\yZ-zY, zX-xZ, xY-yX) 2 = 0. 
And it is to be shown that this is equivalent to the former equation; viz. writing 
yZ — zY, zX — xZ, xY—yX = \, g, v, then that the two equations 
{q'z 2 + r'y 2 , r'x 2 + p'z 2 , p'y 2 + qx 2 , —p'yz, —q'zx, —r'xy§\, g, v) 2 = 0, 
p" X 2 + q'g 2 + r"v 2 — 0, 
are equivalent to each other. 
We have identically \x + gy + vz= 0, and thence also 
(\x + gy + vz) [(p' — q — r') \x +(-p' + q- r') gy + {—p — q' + r') vz] = 0, 
where, on the left-hand side, the terms in gv, v\ and \g are 
= — 2p'yz gv — 2q'zxvX — 2r'xyXg. 
Hence the first equation may be written 
\c[z 2 + r'y 2 + (p' — q— r') x 2 ] \ 2 + \r'x 2 + p'z 2 + (— p + (( — r') y 2 ] g 2 
+ [p'y 2 + q'x 2 + (— p' — q' + /) z 2 ] v 2 = 0, 
and it is to be shown that this is equivalent to 
p"\ 2 + q"g 2 + r"v 2 = 0; 
viz. that we have p" : q" : r" — 
q'z 2 + r’y 2 — ( p' — q' — r') x 2 
: r'x 2 + p'z 2 — (— p' + q' — r') y 2 
: p'y 2 + q'x 2 — (— p' — q' + r') z 2 , 
where px 2 + qy 2 + rz 2 = 0. Writing the equation in the form 
p" : q" : r" = A : B : C, 
54—2