200 
[512 
512. 
ON A CORRESPONDENCE OF POINTS IN RELATION TO TWO 
TETRAHEDRA. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 396—404. Lead June 12, 1873.] 
The following question has been considered by R. Sturm in an interesting paper, 
“ Das Problem der Projectivitat und seine Anwendung auf die Flachen zweiten 
Grades,” Math. Ann., t. I. (1870), pp. 533—574: Given in piano two groups of the 
same number (5, 6, or 7) of points, to find points P, P' homographically related to 
these two groups respectively; viz. the lines from P to the points of the first group 
and those from P' to the points of the second group are to be homographic pencils. 
In the present paper I require only a particular form of these results; viz. in each 
group two of the points are the circular points at infinity; or, disregarding these, we 
have two groups of 3, 4, or 5 points such that the points of the first group at P, 
and those of the second group at P', subtend equal angles. I give for this particular 
case an independent analytical investigation; but I will first state the results included 
in the more general ones obtained by Sturm. 
If the points A, B, C at P and the points A', B!, C' at P' subtend equal angles, 
then to any given position of the one point corresponds a single position of the other 
point; viz. the two points have a (1, 1) correspondence; the nature of this being, that 
to any line in the one figure corresponds in the other figure a quintic curve, having 
6 dps.; viz. the three points, the two circular points at infinity I, J, and one other 
fixed point of that figure (say for the first figure this fixed point is (ABC)}. 
If the points A, B, C, D at P and the points A', B', C', D' at P' subtend equal 
angles, then the locus of each point is a cubic curve; viz. the locus of P passes 
through A, B, C, D, /, J and the four fixed points (ABC), (ABD), (ACD), (BCD); 
and the like for the locus of P'.