298 ME. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SUEFACES. 
If, therefore, 
x, Y, Z, T= 
A, 
B, 
c, 
D 
] 
r, 
s 
u , 
v, 
w, 
k 
V 
P„ P» P 3 , 
A 
B 
C 
D 
+f% 
a 
ft 
7 
1 
t 
Ui 
vo ' 
v' 
V 
a' 
ft' 
7' 
y 
u 
V 
vo 
k 
u 
V 
vo 
p 
) 
(29) will take the form 
(wX-3HP 1 )a+(wY-3HP 2 )h + (%Z-3HP 3 )g+(wT-3HP 4 )l=0, . . (31a) 
to which might be added the analogous equations in h, b, f, m ; g, f, c, n ; 1, m, n, d. 
Another, and for some purposes a more convenient form may be given to these equa- 
tions by the following transformation : 
H=(9T, . . )(A, B, C, D) 2 , suppose; 
d,H=(d,a', • -)(A, B,C, D) 2 +|(a', . .)(A, B, C, D)(«A— «A, (3 A-«B, y A- a C, iA-«D) 
=(W, ..)(A,B, C, B)=+^(Sr, ..)(A,B, C,D)(a,/3, r ,S)-§H. 
If, therefore, henceforward j), q, r, s represent the differential coefficients of H upon 
the supposition that A, B, C, D are regarded as constants, we shall have 
so that 
X=B C D +2H (3 y l 
q r s (3 1 y' h' 
v vo k v vo k 
P,=B C D u [3 y h 
vo' v' 1! (3 1 y' y 
v w k v w k, 
wX— 3HPj= .BCD =(»— 2)- 1 Aa B C D 
3H q r s px q r s 
U W ' V 1 l' —U-\-UyX V)' v' l' 
v vo k ux v vo k; 
and if P, Q, B, S represent the determinants formed from the first three together with 
the fourth, the fifth, the sixth, and the seventh columns respectively of the following 
matrix : 
A 
u 
V 
Ui 
vo ' 
v’ 
11 
B 
V 
9 
vo ' 
u' 
ml 
C 
vo 
r 
v' 
vl 
W l 
vl 
D 
k 
s 
11 
ml 
n! 
h 
