246 MR. W. SPOTTIS W 0 ODE ON MULTIPLE CONTACT OE SUREACES. 
and if the surfaces touch not only at the two points P, P x , but also at a third point 
P 2 , the last equation may be replaced by 
0A| 3 U + 3(2 3)B 23 A 23 U = 0 (12) 
The results of the elimination of the ratios Q : 12, . . will be of the form 
1 B-1 0 A? a U + 3 . O' 1 - 1 ! . 1»" 1 2B 12 A 12 U=0 (13) 
Similarly the five conditions for five-branch contact will be of the form 
(1 b-1 0) 3 A 1 4 2 U-(-6 . P^O . O"" 1 ! . 1 b-1 2B 12 A X2 U + 3(0 B-1 1 . 1 B - ] 2) 2 (A 2 +B 2 )U=0, (14) 
and the six conditions for six-branch contact will be of the form 
(P-'O^ATJ+IO . l^O . 0"" 1 ! . 1 b-1 2BA 3 U + 5(0 b-1 1 . 1*- 1 2) 8 B(A 3 +3B*)U=0. (15) 
Recapitulating the results now obtained, we may form the subjoined Table for the 
possibility of contact of a quadric V with a given surface U, viz. simple contact at a 
point P l5 &c. 
Eor contact at the 
point P. 
Number of 
conditions. 
Degrees of conditions in 
Coefficients of U. 
Coordinates of P. 
Coordinates of P x 
; of P 2 , . . 
3 branch . 
3 
4 
2>1% — o 
n-\- 1 ; 
2 
4 do. . . 
4 
5 
4n—5 
n- j-2 ; 
3 
5 do. 
5 
7 
5n—6 
2n+2; 
4 
6 do. 
6 
8 
6n — 8 
2n+4; 
5 
To these of course must be added the conditions that P and P x lie on the surface 
U, viz. U = 0, U 1 = 0, and that U and V touch at P. 
If the surfaces have simple contact at a third point P 2 , we must add the condition 
U 2 =0, and the condition (18) of § 1 ; and similarly for every additional point at which 
they have simple contact. 
If the surfaces have 3, 4, . . pointic contact at a second, third, . . point we must 
double, triple, . . the number of conditions for each such degree of contact ; the degrees 
of the conditions remaining unchanged. 
Suppose, then, that the quadric V touches the surface U in two points, P, P x ; then in 
order that the contact may become three-pointic at either of these points (say P), three 
conditions are necessary. And if a,b,.. be the coefficients of U, these conditions may 
be expressed thus : 
(a, b, . .)\x,y, . .) Zn -\x„y„ . .) B+1 (# 2 ,y 2 , . .) 3 =0, 
(a, b , . .)\x,y , . .y*~*(x u y u . .) n+1 (x 3 ,y 3 , . .) 2 =0, 
(a, b, . .) 4 (x, y, . .) 3 "- 8 (a? I , . .) n+1 (x„ y 4 , . .) 2 =0 ; 
and if by means of these equations, together with the equation U 1 =0, we eliminate the 
