ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SUEFACES. 
249 
When P is a sextactic point, we have the six additional equations, 
□ * 3 u=o, □ 1,11=0, □ ; 8 u=o, n? 4 u=o, dlu=o, □ 5,11=0. 
From four, or five, of these we may eliminate four, or five, of the quantities Q : 23, 
0 : 31, 6 : 12, Q : 14, Q : 24, 6 : 34, by means of the equations 
□ 2 3 u=o, 0*^=0, □; 8 u=o, □; 4 u=o, n 4 4 u=o, 
and obtain results of the forms 
(BU) 2 A 5 U — 10BU . A 2 U. BA 3 U+5(A 2 U) 2 (A 2 +3B 2 )U = 0, 
(BAU) 2 A 5 U-10BAU . A 5 U . BA 3 U++(A 3 U) 2 (A 2 + 3B 2 )U=0, 
A 4 U, 6BA 2 U, 
A 4 U, 
A 5 U, 10BA 3 U, 
A 5 U, 
3(A 2 +B 2 )U , 
6BA 2 U 
5B(A 2 + 3B 2 )U, 
10BA 3 U, 
3(A 2 +B 2 )U 
5B(A 2 +3B 2 )U 
= 0 . 
If either we have determined all the coefficients of the quadric, or if having deter- 
mined only eight we use one of the last equations for determining the results, we shall 
have twelve conditions. 
Recapitulating, the following is a Table of the number of conditions so found, and of 
their degrees in the several quantities contained in them : — 
No. of 
Degree of condition in the 
For a 
constants 
determined. 
No. of conditions. 
Coordinates 
of P. 
Coordinates 
ofP,.. 
Coefficients 
of U. 
Quartitactic J 
7 
8 
3 
2 
2(3ro-4) 
4 
6 
point . . 1 
9 
1 
Quintactic \ 
point . . 1 
7 
8 
9 
3 + 5 = 8 
2+5=3+4 = 7 
1+5=2 + 4= 3+ 3= 6 
3(3ro-4) 
lift— 16 
6 
5 
9 
11 
Sextactic ( 
point . . 1 
7 
8 
9 
8+6 =14 
7 +6=8+5 =13 
6+6=7+5 = 8+4=12 
2(5ft— 7) 
6(2ft— 3) 
2(9w— 14) 
7 
9 
11 
10 
12 
18 
§ 5. On Multiple Contact of Cubics with other Surfaces. 
Hitherto we have considered in detail only the case of quadrics, that is to say, the 
conditions which must be fulfilled by the coefficients of a surface U in order that it may 
be possible to draw a quadric having contact of given orders with U at more than one 
point. There is of course a corresponding problem with cubics, and indeed with surfaces 
of any degree ; but the question soon becomes so complicated that it may be doubted 
