MR, W. SPOTTIS W 0 ODE ON MULTIPLE CONTACT OF SURFACES. 
255 
of which four groups two only will of course be independent. It appears, therefore, that 
for osculation at three points we have three conditions, and for osculation at four points 
six, when the surface can be touched by a quadric at the four points, instead of two, or 
four, as in the case when such contact is not presupposed. The degree of these condi- 
tions is 13; but one of each three may be depressed to the degree 12 by multiplying 
together the dexter and sinister sides of these expressions and dividing out the common 
factor BB'CCT)D'1” -1 0 . 2” -1 0 . 3' i-1 0, &c. The results, by the help of the relations (17), 
may in the case of osculation at the four points be put into either of the following 
forms, viz. : — 
K'MR'=HO'Q, KM'R=H'OQ', 1 
D'L'R =COP , 
DFQ' =B'KP, 
C'FM =BH'L, 
DLR' = C'0'P, 
D'FQ=BK'P', 
CF'M =B'HL' . , 
(19) 
The arrangement of the letters in these equations will be perhaps more readily 
apprehended by reference to the matrix written below ; by which it appears that the 
combinations, accents apart, follows that of the principal minors of the determinant, 
. , B , C , D, 
F, . H, K, 
L, M, . O, 
P, Q, R, . 
The total conditions will in this case be, four of the degree 1, four of the degree 13, 
and two of the degree 12. 
The form of the above equations will perhaps be best seen by actually writing down 
the first, viz. 
[01'2]2 n_ T + . . [01'3]3’ !_1 1-|-. . 
[02'l]l n-1 2 + . . . [02'3]3 m ~ 1 2+- . 
[03 , 1]P" 1 3 + . . [Q3'2~\2 n -'3+. . 
( 20 ) 
2 N 
MDCCCLXXVI. 
