550 PKOFESSOR CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CTJRYE. 
Article Nos. 9 to 11 . — First Transformation. 
9. We have, assuming always U=0, the following formulae (see post. Article Nos. 31 
to 33):— 
p;+io«d 2 +ioa$ s +i6a 1 aBu 
= { ( 2 7 m 2 — 9 6 m -f- 8 1 ) Hd <E> -fi ( 1 7 m 2 — 5 6 m 5 1 ) $>d H } 
+ ^ I y 4 {(-Um-22)(d.V)H -(10m-18)c>VH} 
d 4 Ud,H + 2d 3 Ud 2 H + 2d 2 Ud 3 H 
=j^^{(-6m 2 +18m-12)H 2 d<P+(-17m 2 +60m-55)II<Pd(P} 
+J~Ij4{(2m-2)H(d . V)H +(8m-16)BHVH} 
(m— I) 4 ^ ^^H}, 
d 2 Ud 3 TJ=- ( ^ ir4 HdH. 
10. And by means of these the condition becomes 
<^2tT2 
0 = ^£ I y 4 {(153m 2 — 594m+549)HBO+(— 102m 2 +396m+366)OBH} 
S 3 H 
+ (^ rT ^{(- 96 m + 168 ) H (B . V)H+(-90m+162)HBVH+(120m-240)BHVH} 
+7^ I p{9 H2 BQ-45HQBH+40^H}, 
being, as already remarked, of the degree 5 in the arbitrary coefficients (X, (m, v), and of 
the order 12m— 22 in the coordinates (x, y, z). 
11. But throwing out the factor ^ 2 , and observing that in the first line the quadric 
functions of m are each a numerical multiple of 51m 2 — 198m-|-183, the condition becomes 
0= (51m 2 -198m+183)H 2 (3HdO-20hH) 
{( — 96m-f 168)H 2 (d . V)H+(-90m+162)H 2 BVH + (120m-240)BHVH} 
+& 2 {9H 2 BO-45HOBH-}-40^H}. 
Article Nos 12 & 13 . — Second transformation. 
12. We effect this by means of the formula 
(m— 2)(3HS$— 2C>BH) = — ^ Jac. (U, C>, H), .... (J)* 
(J) here and elsewhere refers to the Jacobian Formula, see post, Article Nos. 34 & 35. 
