512] ON A CORRESPONDENCE OF POINTS IN RELATION TO TWO TETRAHEDRA. 201 
Finally (although this is a theorem which I do not require), if the points 
A, B, G, D, E at P and the points A', B', G', D', E' at P' subtend equal angles, 
then there are three positions of each point. 
The problem I propose to consider is: Given the tetrahedra ABCD and A'B'C'D', 
it is required in the planes ABC and A'B'G' respectively to find the points P, P' 
such that A, B, G, D at P, and A', B', C', D' at P', subtend equal angles. I was 
led to this by the more general problem, which I do not at present discuss: Given 
the two tetrahedra, it is required to find the loci of the points P, P' such that 
A, B, G, D at P, and A', B', G', D' at P', subtend equal angles. 
Here, drawing from P, B' the perpendiculars JDK, D'K' on the planes ABG and 
A'B'G' respectively, we have A, B, G, K at P, and A', B', G', K' at P', subtending 
equal angles, and such that the distances PK and P'K' are proportional to the heights 
of the tetrahedra (for the triangles PJDK and P'D'K' are obviously similar). The 
required points P, P' are each the intersection of two loci, viz.: 
1. P is such that A, B, C, K at P, and A', B', C', K' at P', subtend equal 
angles; locus is a cubic through A, B, G, K, I, J, (ABG), (ABK), (AGK), 
(BGK). 
2. P is such that A, B, K at P, and A', B', K' at P', subtend equal angles, 
and that PK and P'K' are in a given ratio; locus is a certain octic 
curve il; 
and the required positions of P are obtained as the intersections of the two loci. 
I proceed to the analytical investigation. 
Preliminary Formulae. 
1. Consider a triangle ABG, and let the position of a point P be determined 
by means of its coordinates x, y, z, which are equal to the perpendicular distances of 
P from the sides, each divided by the perpendicular distance of the opposite vertex 
(as usual, x, y, z are positive for a point within the triangle); or what is the same 
thing, x, y, z = PBG, PGA, PAB, divided each by ABG, whence identically x + y-\-z= 1. 
Suppose for a moment the rectangular coordinates of A, B, G are (a ly &), (a 2 , /3 2 ), 
(a 3 , /3 3 ) respectively; and that those of P are X, Y. Also let the sides BC, GA, AB 
be = a, b, c respectively. 
We have 
X = a x x + a 2 y + a s z, 
C. VIII. 
Y = fax + ¡3,y + faz, 
1 = X + y + z\ 
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