MR. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SURFACES. 251 
From these equations we can, by means of the values of the ratios 0 : 0, : 0 2 , eliminate 
0, 0 t , 0 3 , 0 3 in two different ways ; and the resulting equations will be of the degree 
5 + 5 + 8 = 18. 
Proceeding to perform the actual eliminations, and taking the three equations involving 
012, we find that the ratios 
-2. 012 : -0 : 0,: -0 2 
are proportional to the determinants of the matrix, 
O—'l . 0"-% -[O', 1, 2], l n-1 0(0 n-1 2) 2 , 2” _1 0(0 n_I l) 2 , 
l n_1 2 . 1 " _I 0 , o n-1 l(l 7l_1 2 ) 2 , -[ 0 , 1 ', 2 ], 2 "- 1 l(l»- , 0 j 2 , 
2 7l-1 0 . 2“ -1 l, 0"- , 2(2’ l - 1 l) 2 , 1»- I 2(2»- I 0) 2 , -[0,1,2']; 
viz. the quantity to which 0 is proportional will be 
0 B r'l[0, 1', 2]0 B-1 2[0, 1, 2'] 
+ 0 B - J 1[0, 1', 2](2 B-1 0) 2 2 B-1 1 . 0 B-1 1 
+0 B -’2[0, 1, 2'](l n-I 0) 2 0”- I 2 . l B -2 
_ 0 »- i i . o “-‘2 . (1”- 1 0) 2 (2 B - 1 0) 2 1 B - I 2 . 2 B-1 1 
+l B - I 0(0»- 1 l) 2 (l B - 1 2) 2 (2 s - I 0) 2 2 B - , 0 
+ 2”- 1 0(1 b - 1 0) 2 (0 b - , 2) 2 (2 b - 1 1) 2 1 b - 1 0; 
and writing, for brevity, 
2- 1 3 . 3 b_1 1 . l-!2— 3 B ~ X 2 . l n - l 3 . 2 m_1 l=P m , 
2 m_1 3 . 3 B - X 0 . 0 B - x 2-3 m - x 2 . 0 B-X 3 . 2 B_x 0=P o23 , 
l n— >3 . 3 n_! 0 . 0 B-X 1 — 3 B-1 1 . 0 B-X 3 . 1 b - 1 0=P 013 , 
1»-i 2 . 2 B-1 0 . 0 W_X 1 — 2” _1 1 . 0“ _1 2 . 1 b " 1 0=P 012 , 
and forming expressions symmetrical to that written above, we shall find that they may 
all be put into the following shape : — 
0 : 0, : 0 2 
= ([0, 1', 2]0 B - 1 1 + (1»- 1 0) 2 0 B " 1 2 . 1 B-1 2) ([0, 1, 2']0"~ X 2+(2 B-X 0) 2 0 n_1 l . 2"- 1 l) " 
+ 1” _1 0 . 2 b - 1 0P 2 12 
: ([0, 1, 2']1”- 1 2+(2 b - 1 1) 2 1 b - 1 0 . 2 TC " 1 0) ([O', 1, 2]1”- 1 0 + (Q B - 1 1) 2 0 B - X 2 . l"- x 2) ^ ^ 
+2— X 1 . 0 b - x 1P 2 12 
: ([O', 1, 2]2 b_1 0 + (0 n_x 2) 2 2 n ~ x l . 0 B - X l)([0, 1', 2]2”- 1 l+(l B - 1 2) 2 l B - 1 0 . 2— x 0) 
+ 0 »- x 2 . l»-i2P 2 12 J 
Again, from the two equations involving 123, we obtain by elimination 
- ([P, 2, 3]2»- x +(l B - x 2) 2 l B - x 3 . 2»- 1 3)2 b - x 30 1 | 
+ ([1, 2', 3]1 b - x 2+(2»- x 1) 2 1 b - x 3 . 2”- x 3)l B - x 30 2 +l B - x 2 . 2' 1 - 1 1P 123 0 3 =O. f ' ^ 
2 M 
1 
► ( 6 ) 
MDCCCLXXVI. 
