CONTACT AT ANT POINT OP A PLANE CUPVE. 
381 
The coefficients of have a similar form ; and uniting the three terms, ob- 
serving that is equal to 3(m— 2)H, and attending to the values of 
O, □ , we have, restoring the omitted factor 
H«I>+ A5_ 
2 m—l ' m —1 
Proof of the expressions for and D^H 
22. These are obtained (for the particular case m=4, which makes but little differ- 
ence) in Mr. Salmon’s ‘ Higher Plane Curves,’ pp. 88 and 89, and I merely reproduce 
his investigation ; we have 
or, what is the same thing, 
(DH)«={A(Cd,-B3JH+KAB,-C3,)H+<B3,-A3JH}’; 
and if we consider first the term which contains the coefficient is 
{(CB,-BB.)H}1 
Now making use of the equations 
{m~l)A=aa;-]-h;^-{-gz, 
{'in-l)C=cjx+fy-\-cz, 
and 
m{m--l)\]=ax^-\-bf-\-cz'^-{- 2fyz -|- 2 gzx -f- 2 hxy = 0 , 
we have 
{m — l yC^ = (gx -\-fy -\-czy— c{ax^ + + ^fyz + 2 gzx + ^hxy) 
= — 13^^ + 
(m — 1 )^BC = {hx-l- iy +/ ^ ) {gx +fy -f cz) —f{ax^ -}- by^ + cz^ + 2fyz 2 gzx + 21ixy) 
= — (Bxy — ^xz + ^yz, 
{m-iyB'^ ={/ix+by-l-fzy-b{ax^+by^-i-cz^-i-2fyz-j-2gzx-y2/ixy) 
= — -f 20XZ — ^z^ ; 
and hence 
and 
(m-l)^((Cd,-BB,)H)'= {-3Sx^+ 2^xy -^y^){d^Hy 
+^(—Jx^-i-<3xy-l-^xz—^yz)d^II.d^H 
+ {~€x^+ 2(3xz -g[z^){d,uy 
= - JT, CIB^H, 
+2x(yd^H-l-zd,H){^d,H-yed,H) 
yd^R +^B,H = 3(m- 2)H - zd,H, 
3 E 
MDCCCLIX. 
