209 
2G1] 
CONTACT AT ANY POINT OF A PLANE CURVE. 
C. IV. 
27 
the several terms of which must respectively vanish, and we have therefore 
U= 0, 
d,u=o, 
dJJ — — 0/t7, 
d 3 U = -{d^2>dA)U, 
dJJ = - (0/ + 63 1 2 9 2 + 49 x 0 3 + 80,/) U 
5. Next, preparing to substitute in the equation 
IPU-Л. DU = 0, 
the consecutive value of DU is 
(x + dx + \d?x + \d?x + ^d A x) d x U + &c. 
= (0 O + 0i + £0, + ^0з + /4 0 4 ) U, 
where 
0 O U = (хд х + уду + zd z ) U =mU. 
Reducing by the above results, the consecutive value of DU is 
= — £0/Z7— ¡t (0/ + 30!0 2 ) U — /4 (0/ + 60/0 2 + 40! 0 3 + 30/) U. 
C. Hence also writing 
P = ax + by + cz , 
0i P = adx + bdy 4- cdz , 
d.,P — ad 2 x + bd-y -f cd-z, 
the consecutive value of — TIDU is — (P+ d x P + ^d 2 P) multiplied into the consecutive 
value of DU, and the product is 
= P. * Э/£7 
+ P. * (0/ + 3010 2 ) U + дгР. W U 
+ P. (0/ + 60/0, + 40103 + 30/) и+дгР.Ъ(0/ + 30!0 2 ) U + fdD. 10/U 
7. The consecutive value of D-U is 
= (x+dx + %d\x + ^d 3 x + ^d 4 x)- d x 2 U + &c. 
о Л 
= or 
+ 2x dx 
+ x- dx + (dx) 2 
+ $ x d 3 x + dx d 2 x 
+ T \x d*x 4- \dx d 3 x + I (d 2 xf 
-d x 2 U + &c.,