254 
MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SURFACES. 
and similarly for the other groups of points. The degree of these equations, when 
cleared of fractions, is obviously 18, as stated above. 
The requisite conditions are consequently, four of the form 1 B =0, 2”=0, . . , of the 
degrees 1, 1, 1, 1, in the coefficients of U ; and two of the degree 18 in the same quan- 
tities ; six in all. And from these data theorems corresponding to those enunciated 
for contact by a quadric may be written down. 
It is, however, to be noticed that if osculation subsist at four points P, P„ P 2 , P 3 , then 
we have simultaneously the equations 
. 6 X :6^:6 Z — . : BB' + . . : CC' + . . : DD' +. . , ] 
6:.:6 2 :6 ,=EF + . . : . : HH'+ . . : KK'+. . , I . _ _ (17) 
6:6,: . : ^=LL' + . . : MM' + . . : . : 00' , j 
Q:6,:6 2 : . =PP' + ..:QQ' +. . : RR' -f . ) 
any two rows of which being taken as independent the remaining two are consequences. 
Taking any two we can eliminate one of the ratios Q : Q, : : 0 3 , and thus obtain as one of 
the conditions an equation of the degree 16. 
In this case, therefore, there will be four conditions of the degree 1, three of the 
degree 18, and one of the degree 16. 
In certain special cases these expressions undergo considerable modification. Thus, 
if the surface U be capable of being touched by a quadric, as well as being osculated 
by a cubic in the four points P, P,, P 2 , P 3 , we shall have, as proved above, 
P 1 23=0, P 023 =0, P 013 =0, P 0 i2=0 ; 
and the system of conditions for the osculation between the cubic V and the surface U 
will take the following form : — 
K'CC' 3 n-1 0 — HDD'2 n-1 0=r0, 'j ' 
MDD'k-'O— O'BB' 3 n-1 0 = 0, l 
R'BB' 2"- 1 0 — QCC' 1»" I 0 = 0, J 
D'HH'3"~T — CKK' 2 B - 1 1=0, v 
L'KK' O n ~T — OFF' 3’ 1 ~ 1 1=0, t 
BFF 2 n_1 l — P , HH'0 n_1 l = 0, J 
DMM' S 71 -^ — B'OO' l m_1 2=0, ^ 
FOO' 0 n-1 2 — KLL' 3” _1 2=0, l 
QLL' l n - 1 2-PMM'0 w - 1 2=0, J 
C'QQ' 2 W-1 3 — BRR' p->3=0, ^ 
FUR' 0 B_1 3 — HPP' 2 B ~ 1 3=0, l ] 
MPF 1* _1 3 — LQQ' 0 n ~'3=0, J j 
( 18 ) 
