228 ON THE SEXTACTIC POINTS OF A PLANE CURVE. [341 
13. We have (see post, Article Nos. 36 to 40) 
Jac. (U, 3>, if) = -(3.V)tf; 
and introducing also 3. Vif in place of 3 V if by means of the formula 
3Vif = 3(Vif)- (3.V)if, 
the condition becomes 
| 51 m 2 —l$8m + i83 _ (0 v ) # 
+ (- 90m + 162) m (V if) +120 (m - 2) HdH V if 
+ * (9if 2 3il - 45ifil3if + 40^3if) = 0, 
or, as this may be written, 
(45m 2 - 180m + 171) if 2 (3.V) H 
+ (- 90m + 162) (m - 2) if 2 3( V if) + 120 (m - 2) 2 if3iiV if 
+ (m - 2) * (9if 2 3-Q - 45ifH3if + 40^3if) = 0. 
Article Nos. 14 to 17.—Third transformation. 
14. We have the following formulae, 
A Jac. (U, Vif, if) -(5m-ll)3//Vif+(3m-6)if3(Vif) =0, (J) 
A Jac. (U, V, if) if — (2m — 4)3ifVH + (3m-6)H (3.V)if=0, (J) 
in the latter of which, treating V as a function of the coordinates, we first form the 
symbol Jac. (U, V, if), and then operating therewith on H, we have Jac. (U, V, if) if; 
these give 
m < v ^ dHyI! ~ 3(^--2)< P - 
H(d.V)H= (i/, V , 
and substituting these values, the resulting coefficient of if3ifV if is 
(45m 2 — 180m + 171) § 
+ (- 90m+162) 5m ~ 11 
which is = 0. 
+ 120 (m - 2) 2 ,