136 
PROFESSOR CAYLEY ON THE CURVES 
viz. this is 
viz. in the rational form this is 
360 2 U— x , y , z 2 =0, 
n/0, 1, 0 , 0 3 
(x/0)' . 1, 34 2 
(x/0)" . . 60 
and this will have at the point (1, 0 , 0 3 ) a contact of the third order if 0 be determined by 
x/0, 1,0, 0 3 =0, 
(x/0)' . 1, 30 2 
(x/0)" . . 60 
(x/©)'" . . 6 
0(x/0) w -(x/0) 7 =O; 
or developing and multiplying by 0^, this is 
0 { @ 2 0'" _ f©©'©" -f |@' 3 } — (@ 2 ©" — i©©' 2 ) — 0, 
or, what is the same thing, 
© 2 (40"'_0") 
+ 00'(--fS0" + ±0') 
+ 0' 2 .fS0'=O; 
and substituting for 0 its value, this is 
(cS 6 + 2/S 4 + 2^9 3 +5S 2 -|- 2AS + a) 2 (45cS 4 + 12/S 2 — b) 
+ (cS 6 + 2/S 4 + 2</S 3 + U 2 + 2 M+a) (3 cS 5 + 4/S 3 + 3#S 2 + bb + h)( -42cS 5 - 32/S 3 
— 15^0 2 — 2SS+A) 
+ 3S( 3cS 5 + 4/S 3 + 3^S 2 + 66 + 7i) 3 = 0. 
The coefficients of the powers 16, 15, 14, 13 of S all vanish, so that this is in fact an 
equation of the twelfth order (*/0, 1) 12 =Q; and putting, as usual, 
(bc-f 2 , ca-g 2 , ab-li 2 , gh-af, hf-bg,fg-ch)=(A, B, C, F, G, H), 
the equation is found to be 
- 4cA 
S 12 
-1- 72AA 'j 
+ 45AB v 
+ 30cH 
S 11 
+ 
>S 7 
-20/C Is 4 
— 36cB 
j 
- 227>H J 
+ 10AGJ 
+16/A 
+ 40aA | 
+ 5AF I 
'S 3 
-lOcG 
Js® 
— 130AH 
+ 20aGj 
+ 40^A 
+ 10yG 
rO 
- UC 1 
S 2 
-12aF J 
+ 20SA 
) 
+ 40/F 
-60/B 
Is 8 
+ 337iB > 
I 
1 
Ox 
o 
Q> 
— 90/H 
J 
+ 2SG 
— 108«H j 
's 5 
— a C =< 
