512] 
IN RELATION TO TWO TETRAHEJDRA. 
207 
7. Now the locus of P is evidently a curve, and this can only happen by reason 
that the four left-hand functions contain a common factor, and the form of them 
suggests that this common factor is ABCDOOfifiJiJ, the four extraneous factors being 
P 2 / 2 / 2 , A 2 / 2 «/ 2 , B 2 I 2 J 2 , G 2 I 2 J 2 ; viz. ABGB00 1 0 2 0 3 IJ =0 is a cubic curve passing through 
the ten points; and D 2 I 2 J 2 = 0 a cubic curve through each of the points D, I, J 
twice; viz. it is the triad of lines IJ, DI, DJ; and the like as to the other 
extraneous factors A 2 I 2 J 2 , B 2 I 2 J 2 , and C 2 I 2 J 2 . I have not worked out the analysis to 
verify this d posteriori; but, the conclusion agreeing with Sturm, I accept it without 
further investigation, viz. the result is that A, B, G, D at P and A', B', G', D' at P' 
subtending equal angles, the locus of P is a cubic curve ABGD00 1 0 2 0 s 0 4 IJ = 0 
through the ten points thus represented; and of course the locus of P' is in like 
manner a cubic curve A'B'C'B'0'0 1 '0 2 0 3 'O i 'IJ=0 through the ten points thus represented. 
Correspondence, A, B, C, I), E at P and A', B', G', D', E' at P' subtending equal angles. 
8. We may go a step further, and consider A, B, G, D, E at P and A', B', C', D', E' 
at P' subtending equal angles. Attending only to the points A, B, C, D and A', B', G', D', 
the locus of P is a cubic curve 
ABCD00 1 0 2 0 3 0JJ = 0; 
and similarly attending to the points A, B, G, E and A', B', G', E', the locus of P 
is a cubic curve 
ABGEOQ 1 Q 2 Q 3 IJ= 0. 
(Observe that 0, as depending only on A, B, G, is the same point as before; but 
that Q 1} Q 2 , Q 3 , as depending on E instead of JD, are not the same as 0 lt 0 2 , 0 3 .) 
The two cubic curves have in common the points A, B, G, I, J, 0, and they con 
sequently intersect in three other points; that is, there are three positions of the 
point P, and of course three corresponding positions of P'. 
Correspondence, A, B, G at P and A', B', C' at P' subtending equal angles, and AP, A'P' 
in a given ratio. 
9. Consider, as before, A, B, G at P and A', B', G' at P' subtending equal angles, 
and the points P, P' being moreover such that the distances AP, A'P' are in a given 
ratio. I write for shortness 
where L, M, N denote 
V z 
: M : N’ 
U + A (cot A — cot A') fix, U + A (cot B — cot B') £ly, V -f A (cot G — cot G') Oz, 
respectively. We have 
(AP) 2 = — a 2 yz — b 2 z(x — 1) — c 2 (x— 1 )y, 
= — a 2 yz — b 2 zx — c 2 xy + (b 2 z + c 2 y) (x + y + z), 
_ c 2 y 2 + b 2 z 2 + (b 2 + c 2 — a 2 ) yz;