CONTACT AT ANY POINT OF A PLANE CURVE. 
377 
the values of Eg, E 3 , E 4 being 
■i 
E,=PI, 
K3 =dh, 
E4=I)^h 
m~l 
□ 
3{m — 2) 
m — 1 
15. We have Ea^H, and it will be shown that 
+ 3(m— 2)Hn^ 
- 
where for shortness 
and hence Avriting 
we have 
(m-l)^E4=-12(m-2fHO 
+ 4(m— 2) 
- 
B, C, #, #, B,H/, 
a = (^, 13 , c, #, a, II )^H, 
A = 3E2(«i- 1)% - 4(w- 1 )"EI, 
9ffA=-3nH+4-^; 
or replacing O, T' by their values, 
jr, .3, .3. rn.Hi 
-SH» I ^ 3,H)> 1 ’ 
and A having this value, the equation of the five-pointic conic is 
D^U- (f g DH+ ADU) DU=0, 
where it will be recollected that the current coordinates are (X, Y, Z), and that D 
denotes 
16. I remark, in passing, that the problem of finding the circle of curvature at a given 
point of a plane curve, is in fact that of determining the conic having with the curve 
at the given point a three-pointic contact, and besides passing through two given points. 
The equation of a conic having an ordinary contact, is 
D^U-nDU = 0 , 
where 
n = (zX -j“ ^ Y -f” cZ, 
and the condition of a three-pointic contact is 
ax-\-by-{-cz-=2(m—2). 
Let the coordinates of the two given points be 
(•^ 1 , (^ 2 , 3 / 2 , ^ 2 ), 
