ME. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SUEFACES. 301 
Next it may be shown that H is a factor of this expression. For when 11 = 0 the 
following equations subsist, 
Ax+Ry + Cz + 1)£=0 
px-\ -qy -\-rz +st =0 
ux-\-vy -\-wz+kt =0, 
whence by elimination 
X_Y_ZT_ P 
x y z t ( n — \)u ’ 
so that, omitting a numerical factor, uX—x P will be replaced by #P, and (36) will then 
become 
*B,P+P 
^P 
^yP + P 
a'B.P 
a-B<P 
zc^P A u 
sByP B v 
zB.P + P C w 
zd t F D k 
ffAP + 2(3B,+ ..)P yAP+2($B,+ . .)P zAP+2(<gB,+ ..)P 
But by adding x (line 1 )-\-y (line 2 )-\-z (line 3) to t (line 4), the whole of line 4 will be 
divisible by P and a numerical factor ; so that the expression becomes 
a’BJP + P 2 /B x P z*BP A u 
aByP ?/ByP + P sByP B v 
#B*P yb s P zB s P+P C w 
x y z . . 
*AP+2(0B,+ ..)P yAP+2(lB,+ ..)P 2 AP+2(<aB,+ ..)P 
Again, subtracting B^P line 4 from line 1 
B y P line 4 from line 2 
B JP line 4 from line 3 
AP line 4 from line 5, 
and dividing throughout by 2, the expression 
is reduced to 
P 
. 
A u 
P 
. 
B v 
• 
P 
C w 
x y 
z 
• 
(9B...)P »B # +..)P 
(<ga.+ ..)P 
• 
lastly, subtracting 
A (column 1) -f-B (column 2) 
and 
u (column 1) (column 2) 
+ C (column 3) from P (column 4), 
-f-w (column 3) from P (column 5), 
