250 
MR. W. SPOTTISW OODE ON MULTIPLE CONTACT OP SURFACES. 
whether the results would be worth having even if means were found for pushing the 
investigation much further. There is, however, one case, namely the osculation of 
cubics, in which it is possible within a moderate compass to arrive at a solution. 
Thus, when V is a cubic, 
□ 12 U=0 2 2 . 0” -1 l— 0 2 1 . 0 ra-1 2, v 
□ ? 2 U= (0 2 2) 2 0”- 2 l 2 - 2 . 0 2 2 . 0 2 1 . 0" _2 12 +-(0 2 l) 2 0”- 2 2 2 t . . (1) 
-0 2 1 . 02 2 . O^-'l-HO^ . 0"-T + 0 2 l . 0" _1 2)012 — 0 2 2 . 01 2 . (V-^J 
But in the same way, as in the case of quadrics (22), we may put 
0 2 1=S0”- 1 1, 0 2 2=3 0"- 1 2, | ^ 
1 2 0=S 1 1»- 1 0, 2 2 0=$ 2 2" -1 0 -J * 
and the conditions for osculation at the point P, viz. □ 2 3 U=0, □ 2 1 U=0, □ 2 2 U=0, will 
then become 
2 . 0” -, 2 . 0 n_1 3 . 023= — ^[0', 2, 3] . + 0 2 2 n -ffi(O n - 1 3) 2 + 0 3 3”-ffi(O’ l - 1 2)V| 
2.0 re - 1 3.0”-T.031 = -^[0 / , 3, l]+^l re-1 0(0" -1 3) 2 + . +4 S 3”- 1 0(0' 1 - 1 1) 2 ,[(3) 
2 . 0 n -'l . Q n -'2 . 012 = -0[O', 1, 2]+^l”- 1 0(0 n-1 2) 2 + ^ 2 2” _I 0(0" _1 l) 2 + . J 
Similarly for osculation at a second point P, we should have 
2 . l*"^ . l^O . 13O=0O” -1 l(P -, 3) 2 — ^[P, 3, 0] . +^3’ l -T(l“- 1 0) 2 , v 
2 . P-'O . p- J 2 . 1O2=0O" - T(P -1 2) 2 — ^[P, 0, 2] + 0 2 2»-T(p- 1 O) 2 . t (4) 
2 . P -I 2 . l-*3 .123= . -4[P, 2, 3]+0 2 2„_ 1 l(p- 1 3) 2 +^3»- 1 l(p- 1 2) 2 ;J 
and for osculation at a third point P„ we should have 
2 . 2” _1 0 . 2’ 1 "T . 201 =50 51 - I 2(2’ l -T) 2 +^ 1 P- I 2(2 ft - 1 0) 2 — 0 2 [2', 0, 1] . v 
2 . 2 n -T . 2“- 1 3 . 213= . +^ l l”- 1 2(2’ l - 1 3) 2 -^[2', 1, 3]+^3 n - 1 2(2’ 1 - 1 l) 2 ,l (5) 
2 . 2™- 1 3 . 2 ,l_1 0 . 23O=0O" _I 2(2 ,l ~ 1 3) 2 . -0 a [2', 3, 0] + ^3”- 1 2(2’ l - 1 0) 2 J 
These together form a system of nine equations involving the seven quantities 
123 : 230 : 301 : 012 : $ : : d 2 : d 3 . We can therefore eliminate these quantities in 
two different ways ; in other words, there will be two relations between the coefficients 
of the equation of the surface U. In order to determine the degrees of these resultants 
in the coefficients in question, consider first the three equations in 012, 0, 0 n 0 2 . The 
degrees of the coefficients of these quantities (in the coefficients of U) are 2, 3, 3, 3 
respectively ; hence the quantities Q, Q 1} 0. 2 will be proportionate to expressions which 
are of the degrees 3 + 3 + 2 = 8. Next taking the two equations involving 123, and 
eliminating that quantity between them, we shall obtain an equation in 0 2 , 0 3 , the 
coefficients of which are of the degree 5 in those of U. Similarly from the others we 
should obtain equations in 0, 0 X , 0 2 ; 0, 0 J5 Q 3 ; whose coefficients are of the same degree. 
