PROFESS OK CAYLEY ON THE SEXTACTIC POINTS OE A PLANE CURVE. 551 
for substituting this value of (3Hb<E>— 205 H) the equation becomes divisible by SJ ; 
and dividing out accordingly, the condition becomes 
51m 2 — 198m + 183 
m — 2 
H 2 Jac. (U, O, H) 
+(— 96m+l 6 8)H 2 (b . V)H+(-90m+162)H 2 bVH+(120m-240)HbHVH 
+^(9H 2 b0-45Ii05H + 40'IbH)=0. 
13. We have (see post, Article Nos. 36 to 40) 
Jac. (U, $, H)=— (d. V)H; 
and introducing also 5 . VH in place of b VII by means of the formula 
bVH=b(VH)-(b.V)H, 
the condition becomes 
|5Irf -198», + l83 _ (6m _ 6) | H8(3 _ y )H 
+ (_90m+162)H 2 b(VH) +120(m-2)HbHVH 
+a(9H a bQ-45HQbH+4(WbH)=0, 
or, as this may be written, 
(45m 2 -180m+171)H 2 (b . V)H 
+(— 90to + 162)(to— 2)H 2 b(VH)+12Q(m— 2) 2 HbHVH 
+(m-2)S-(9H 2 bO-45HQbH+40^bH)=0. 
Article Nos. 14 to 17 . — Third transformation. 
14. We have the following formulae, 
^Jac.(U, VH, H)— (5m— 11)BHVH+(3 to— 6)Hb(VH)=0, . ... (J) 
S-Jac. (U, V, H)H— (2m— 4 )bHVH+(3m— 6)H(b . V)H=0, . . . . (J) 
in the latter of which, treating V as a function of the coordinates, we first form the 
symbol Jac. (U, V, II), and then operating therewith on H, we have Jac. (U, V, H)H ; 
these give 
Jac. (U, VH, H), 
H(3.V)H= f9HVH-3jA^jJac.(U,V , H)H; 
and substituting these values, the resulting coefficient of HbHVH is 
( 45m 2 -180m+171)f 
+ (-90m+162)^=^ 
+ 120( m—2) 2 , 
which is =0. 
