ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEFACES. 
231 
or the points P, P l5 P 3 , giving 
0 ,i-1 l . P 1 "^ . 3 ,l_1 0=l” _1 0 . 3 n_1 l . 0 n ~ lc 6 (21) 
But since in any one of these conditions (19), (20), (21) combined with (18) will 
ensure contact at the four points, it follows that the four equations (18) . . (21) must 
be equivalent to only two independent relations. It is perhaps worth while to verify 
this analytically. In fact, if we multiply together the dexter and sinister sides of (19), 
(20) , (21) respectively, and reject the common factor 
k=0 n ~^ . l—^ . 2"- 1 3 . S^O . 3 ,1-1 1 . 
we shall reproduce the equation (18). Again, if we represent the equations (19), (20), 
(21) , (18) respectively by the following expressions, 
a=a', b—b\ c—d, cl=d', 
the multiplications above indicated give as their result 
dbc=a'Vd=kd=M ; 
and consequently if any two of the equations a=a', b=b', c~d be satisfied, the third, 
as well as the equation d=d', will be so also ; which is the verification required. 
This, as Professor Cayley has pointed out, gives rise to an interesting theorem of 
Solid Geometry ; viz. writing for greater symmetry a, $, y, S, instead of P, P l5 P 2 , P 3 , and 
considering ABCD the tetrahedron formed by the tangent planes of U at the same four 
points respectively, we have the tetrahedron ABCD, the planes whereof contain the 
points a, (3, y, respectively. Now the plane of a0y determines with the planes BCD, 
CDA, DAB, a triangle, the sides whereof contain the points a, (3, y, and which is such 
that the lines joining the summits with the opposite points meet in a point. Similarly 
the planes of /3y5, y£«, determine three other triangles having similar properties. 
And the theorem is that, if the foregoing relation be satisfied for any two of the tri- 
angles, it will be satisfied for the other two. 
The equation (18) may be regarded either as a relation between the coordinates of 
the three points P, Pj, P 2 , or as a relation between the coefficients of U. In the first 
case it shows that if two of the points be taken arbitrarily, the third must be on a 
curve defined by (18) together with the equation U=0. As to the fourth point of 
contact, if such there be, suppose that we represent the equations (19), (20), (21), (18) 
by the symbols (1, 2, 3) = 0, (0, 2, 3) = 0, (0, 1, 3) = 0, (0, 1, 2) = 0; then taking arbi- 
trarily the points P, P l5 the curve on which P 2 must lie will be given by the equation 
(0, 1, 2) = 0. The equation (0, 1, 3)=0 merely shows that P 3 must lie on the same 
curve; but the equation (0, 2, 3) shows that, if P 2 be taken arbitrarily on the curve in 
question, P 3 will lie at one of a finite number of points on the curve. In fact, if we 
take U 3 =0 as the condition that P 3 shall lie on the surface, (0, 1, 3) = 0 as the equa- 
tion expressing the condition that P 3 shall lie on the curve, and (0, 2, 3) the addi- 
tional condition for contact at P 3 , we shall have three equations each of the degree n 
for determining the coordinates of P 3 . The number of positions for P 3 will therefore 
