296 ME. W. SPOTTISWOODE ON THE CONTACT OE CONICS WITH SUEFACES. 
also that 
And farther, if we write 
AU=4H. 
c^H =p, cLFI=r, B/H=s, 
( 22 ) 
and if we suppose that the first four columns in the four following determinants are the 
same as the first four in the expression for H given in (19), then 
Au —p—2 . . 0, 
II 
<1 
A 
CO 
1 
II 
< 
r—2 . . b, Ah 
• = s— 2 
-/ 
. . c 
.. 0 
. . — a 
. .-b 
. . a 
... 0 
.—h 
•• / 
■ • g 
. . h 
. 
. 0 
. . 0 
. . 0 
. . 0 
- 0, 
from which it follows 
that 
A( vh -wg-\-ka)~ 
qh — rg 
A ( wf — nh -\-](b) = 
rf — ph-\-sb 
. . 
A( ug-vf+Jcc) = 
pg-if+sc 
A( — ua—vb — wc) — - 
1 
bo 
1 
1 
c$ 
(23) 
( 24 ) 
the outstanding terms which occur in (23) having cancelled one another. 
Furthermore, applying the formula (21) to the following products, and bearing in 
mind that ch.V, <^V, . . are all linear in x, y , z , t , we have, by means of (24), 
Ad ,V(wA — ivg-\-ka)='d x Y{ qh — rg +sa)+2H( h ^V-ff'b&V-had&Y) 
AbyV {wf— uh + kb )= V(r/‘ — yh -f - sb ) + 2 FI ( fby& . V — 
■libybyV + &d/fV), r 
. (25) 
so that 
A { bfY {vh —wg -f- ka) -f- ~bfs{wf - — uh + kb) -j -..}=« 
p d,V . . 
. (26) 
ft 1 <1 a,v 
7 
y r b,Y 
l 
y s a,v. 
the coefficients of 2H in (25) having cancelled one another. 
This being premised, let us return to the equations (20). These equations being per- 
fectly general, we may replace V by any other function we please. Flence if Ave replace 
V by □ V, the system resulting will be an equivalent for the system □Y = 0, □ 2 V = 0, 
□ 3 V = 0, viz. the second, third, and fourth of the system (8). Making this substitution 
in (20), and remembering that mV is of the degree n, so that the numerical factor 
2 On- 11 
■ 3, 
we have 
n — 1 
