232 
MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OE SURE ACES. 
apparently be n 3 ; but this number doubtless admits of some reduction in consequence of 
the particular form of the equations. In fact, if «, b . . be the coefficients of the highest 
powers of % 3 , y 3 , . . in the equation 3 W = 0, then the terms involving a 3 , b 3 , . . in (18) all 
vanish. But I have not as yet fully investigated this question. 
Turning to the second view of the case, which is in fact the more interesting, viz. 
that in which (18) is to be considered as a relation between the coefficients of U, we 
have for every point of contact four equations, viz. 
U=0, bJJ : 3,U : b z U=b,Y : B,Y : b a Y, V=0, 
or their equivalents ; i. e. for jp points of contact 4j> equations, viz. 
jp equations involving the coefficients of U alone, 
2jp „ „ U & Y, 
jp „ „ V alone. 
But of the 2 p equations which involve the coefficients of U & Y, p—2 may be cleared 
of those belonging to V, and reduced to the form (18) ; and we hence have finally 
2jp — 2 equations involving the coefficients of U alone, 
P+1 „ „ U & Y, 
jp ,, „ Y alone. 
By means of the last 2p-\-2 equations let us determine as many as possible of the 
coefficients of Y. Then if 2p+2>9 we shall be able to eliminate those coefficients in 
2p — 7 different ways, and obtain 2p — 7 equations involving the coefficients of U to the 
degree 10— p ; p being supposed >-9. We may thence form the following table of 
conditions to which the surface U will be subject: — 
Number 
of points. 
Number of 
coefficients of 
V determined. 
Number of 
conditions 
in U. 
Numbers and degrees of conditions in 
coefficients of U. 
Of degree 1. 
Of degree 3. 
Also 
3 
8 
4 
3 
1 
4 
9 
7 
4 
2 
1 of degree 6 
5 
9 
11 
5 
3 
3 „ „ 
5 
6 
9 
15 
6 
4 
5 „ „ 
4 
7 
9 
19 
7 
5 
7 „ „ 
3 
8 
9 
23 
8 
6 
9 „ „ 
2 
9 
9 
27 
9 
7 
11 „ „ 
1 
It must, however, be owned that these numbers, although doubtless true as superior 
limits, must evidently undergo some reduction after a detailed examination of the 
equations upon which they depend. And on this account I abstain at present from 
writing down the geometrical theorems which will obviously occur to the reader on 
perusing these results. 
Thus far for simple or two pointic contact. In order to find the conditions for 
three-branch contact at several points, we must, as in the second paper above quoted, 
