ME. W. SPOTTISW 0 ODE ON THE CONTACT OE SUEFACES. 
270 
where 
U V w k 
which agree with the results obtained in the memoir. 
In order to determine a quadric V which shall have superficial four-pointic contact 
with the surface U, let H u be the value of H when a, 3 , 7, & are written for A, B, C, D 
respectively in the last line and the last column of II ; let H I2 be the value of H when 
a, 3 , 7, c) are written for A, B, C, D respectively in the last line or the last column, and 
a', 3 ', 7', l' in the last column or the last line of II ; and let H 22 be the value of II 
when a!, ft, 7', V are written for A, B, C, D respectively in the last line and the last 
column of H. Further, let 
=p, =q, 3.H=r, b,H=s, 
(58) 
the differentiations being effected (as was shown in the memoir to be permissible) 
without reference to A, B, C, D. Lastly, if p n , . ., yq 2 , . ., . . be the values of p , . . 
when H becomes H n , H I2 , H 22 , respectively, let 
, Y, Z, T= 
A, 
B, 
C, D, 
, P, Q, B, S= — 
A, 
u, 
V > 
w„ 
w' , 
v' , 
V , 
u. 
v , 
iv , Jc , 
B, 
V , 
<L 
iv 1 , 
id , 
in ' , 
ih 
2. 
r , s , 
c, 
w, 
r, 
v' , 
v ! , 
u' , 
D, 
Jc, 
s, 
V , 
m', 
u ' , 
K- 
That is to say, X, Y, Z, T are the determinants formed from the matrix opposite to 
them by omitting each of the columns in order ; and P, Q, B, S are the negatives of the 
determinants formed from the matrix opposite to them by omitting each of the columns 
4, 5, 6, 7 in order, and always retaining the columns 1, 2, 3. 
This being premised, the conditions which the coefficients of the quadric 
V=(a, b, c, d, f, g, h, 1, m, n)(ar, y, z, tf (60) 
must satisfy in order that four-pointic contact may subsist between the two curves of 
section of the surfaces U, V made by the plane Aa’ + B^-f-C^-j-D^O will be, as proved 
in the memoir above quoted, 
(uX -xP)& + (uY — 3 /P)h +(uZ-zP)g+(uT-tP)l =0, 
(vX -xQ)h + (vY -i/Q)b +(vZ -zQ)f + (vT -tQ)m=0, 
(wX-x R)g+(wY-yR)f + (wZ-zB)c+(wT-£B)n=0, [ ‘ ‘ (G1) 
(JcX - xS )1 + (IcY - 3 /S )m +(/rZ- ^S )n + (kT - tS )d = 0 ; j 
and the contact will be circumaxal if the foregoing equations are made independent of 
m' : If, therefore, we represent by X IU , . . P m , . . ; X U2 , . . P 112 , . . ; X 122 , . . P 122 , . . ; 
X 222 , . . P 222J . . the coefficients of the powers of ns' : ns in X, . . P, . . respectively, we shall 
have four equations in the place of each one of the above group — apparently twelve 
