234 
ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SUEEACES. 
We also obtain 
(0 = ) 
0”-' l 
01 
_0”- 1 2_0“-'3 
“ 02 — 03 
2.0 n ~ 1 2.0 n - 1 3.23 
[O', 2, 3] 
2 . 0 n_1 3 . 0 re_1 l . 31 
[O', 3, 1] 
2 . 0 m_ 1 1 . 0 re_ 1 2 . 1 2 
[O', 1, 2] : 
which are the five conditions in order that U, assumed to pass through P, and also 
through P„ P 2 , P 3 , may have a three-branch contact at P. 
Similarly for three-branched contact at the point P, we should find 
30 : 02 : 23=1- , 2[T S 3, 0] : 0, 2] : l-’Op.', 2, 3], 
and for osculation at the point P 2 , 
30 : 01 : 13=2 Jl ' 1 l[2', 30] : 2”- 1 3[2', 0, 1] : 2”" 1 0[2', 1, 3]. 
Substituting from these equations the values of 30 : 23, 13 : 30, 23 : 31 in the iden- 
tical equation 
(30 : 23)(13 : 30)(23 : 31)=1, 
we obtain the following relation, 
l B-1 2[r, 3, 0]2” _1 0[2', 1, 3]0”~T[0', 2, 3] 
=1”- 1 0[1', 2, 3]2" _1 1[2', 3, 0]0" -1 2[0', 3, 1], 
which, in virtue of the condition (18), may be reduced to the form 
[O', 2, 3][1', 3, 0][2', 1, 3]=[0', 3, 1][1', 2, 3][2', 3, 0] (26) 
This therefore is a condition which, in addition to those found before, must be ful- 
filled in order that it may be possible to draw through four points P, P,, P 2 , P 3 a 
quadric surface which will touch the surface U in those points, and have three-branched 
contact with it in three of them. 
The total number of conditions for two-branched contact at four points, and three- 
branched at one of them, may be calculated as follows : — For three-branched contact 
at three points we shall have the following numbers of equations in the coefficients : 
3 equations in U of the degree 1, 
1 55 55 8 , 
1 9 
- 1 * 33 3 3 ^5 
5 „ U and V of the degrees in U, 1 ; in V, 1, 
55 55 55 1 5 55 ^5 
3 „ V of the degree 1. 
