208 ON A CORRESPONDENCE OF POINTS IN RELATION TO TWO TETRAHEDRA. [512 
d 2 y' 2 + b' 2 z' 2 4- 2Ve' cos A'. y'z _ 0., c 2 y 2 + b 2 z 2 + 2be cos A . yz _ 
(a/ + y' + /) 2 (« + 2/ + f) 2 
viz. substituting for x', y', z' their values, this is 
L 2 (x + y + zf (b' 2 z 2 M 2 + c' 2 y 2 N 2 + 2b'c'yzMN cos A) 
= 0 2 (b 2 z 2 + c 2 y 2 + 2bcyz cos J.) {xMN + ylVX + zLM) 2 , 
which is an equation of the 12th order. I say that the points A, B, G, 0, I, J are 
each quadruple. In fact, according to the foregoing algorithm, we may write 
x + y + z = IJ, zM = AB . CA 01 J, &c., 
xL = yM = zN= A 2 BG0IJ, 
y — z = A, xMN = BG. CA 01 J. ABOIJ, &c., 
xMN= yNL = zLM = (.ABGOIJf ; 
and the equation is 
that is 
(.BCOIJ) 2 (IJ) 2 (A 2 BC0IJy = 6 2 . A 2 (ABCOIJy, 
(IJ) 2 (ABGOIJ f = 6 2 . A 2 (ABGOIJf ; 
so that the points are each quadruple. 
The two Tetrahedra ; A, B, G, D at P in ABC and A', B', G', D' at P' in A'B'G' 
subtending equal angles. 
10. I consider now the before-mentioned problem of the two tetrahedra; viz. on the 
two bases ABC and A'B'C' respectively, letting fall the perpendiculars DK and D'K', 
then first A, B, G, K at P and A', B', G', K' at P' subtend equal angles; the locus 
of P is a cubic curve ABGKOOJOJOJiJ = 0 through these ten points. (0 = ABC is 
derived from the points A, B, C; and in like manner 0 1 = BGK, 0 2 — CAK, 0 3 = ABK.) 
Next, B, C, K at P and B', C', K' at P' subtend equal angles, and moreover the 
distances KP and K'P' are in a given ratio; the locus of P is a 12-thic curve 
(BGKOJJf = 0, 
having each of these six points as a quadruple point. Hence among the 86 inter 
sections of the two curves we have the points B, C, K, 0 1} I, J each 4 times, and 
there remain 36 — 24, =12 intersections. 
The conclusion is that A, B, G, D at a point P of ABC, and A', B', G', D' at 
a point P' of A'B'G', subtending equal angles, there are 12 positions of P, and of 
course 12 corresponding positions of P'.