230 
ME. W. SPOTTISW OODE ON MULTIPLE CONTACT OF SUEFACES. 
In the same way the conditions for osculation at a second point Pj will be comprised 
in the system 
□* 2 u 1= o, □*,u 1 =o, ... □ 02 n 03 u 1 =o, ( 16 ) 
and so on for any number of points. 
Returning to the equations (11), (14), (15), which express the conditions for contact 
at the three points P, P n P 2 , and selecting a suitable member from each system, we 
may form the following group : 
□ 12 U=0, CUU^O, □oiU 2 =0 (17) 
These written in full are as follows : — 
01 . O’ 1 - 1 2=02 . 0” _1 1 
12 . I” -1 0=10 . l m-1 2 
20 . 2 n ~ 1 1=21 . 2” -1 0, 
whence, multiplying together the dexter and sinister sides of these equations respec- 
tively, and rejecting the common factor 12 . 20 . 01, we obtain as a condition for the 
possibility of a quadric V touching a surface U in the three points P, P I5 P 2 , the 
following equation : — 
Qn-i j m ln -i 2 o 2 »-' 0=0"- 1 2 . I” -1 0 . 2 n ~ x 1 (18) 
This equation shows that the three points must be so situated that each lies on one 
of the intersections of the first polars of the other two with respect to the surface U ; 
for it may be written in each of the following forms, viz. 
0 ,i -‘l-x0 re - 1 2 = 0, 
l"- 1 2—^1”-’ 0 = 0, 
2 n_1 0— v 2” _1 1 = 0. 
In order to account geometrically for the existence of the equation of condition, we 
may observe, as Professor Cayley has remarked, that, drawing through the points 
P, P 15 P 2 a plane, this plane meets the three tangent-planes of the surface U in the 
three sides of a triangle, which sides pass respectively through the points P, P„ P 2 ; and 
also meets the surface V in a conic touching the sides of the triangle in these points 
P, P l5 P 2 respectively. Considering these points as given, a relation is implied between 
the directions of the three sides ; viz. the triangle must be such that if each summit be 
joined to the opposite point the three joining lines will meet in a point. 
In order that contact may subsist for a fourth point P 3 we may take either the 
points Pj, P 2 , P 3 , giving as the condition 
I*" 1 2 . 2*- 1 3 . 3” _1 1=2" -1 1 . 3 m_1 2 . l n_1 3 (19) 
or the points P 2 , P, P 3 , giving 
2” -1 0 . 0"- 1 3 . 3 W-1 2=0 W_1 2 . fr-'O . 2"~ 1 3 . 
• (20) 
