CONTACT AT ANY POINT OP A PLANE CUEVE. 
399 
which will be divisible by DU if — nD"U+fD®U is divisible by DU, and the condition 
for this is found to be, as before, 
n=|^DH+ADU, 
where A is arbitrary ; we have thus the conditions of a four-pointic contact. 
55. Substituting this value of 11, we see that D'U~^ DH.D'*U divides by DU, viz. 
there exists an identical equation, 
D^U - g DH . D^U = lU + J . DU ; 
and hence if U = 0, 
du(d’u-hDH-D’u)=j, 
where J is a quadric function of (X, Y, Z). I do not know the general form of this 
function, but Mr. Salmon has obtained a result which may be generalized as follows, 
viz. writing for X, Y, Z the values Ca — Av, A/a — BX (where, as before. A, B, C 
are the first derived functions of U and X, f/j, v are arbitrary), the expression for J is 
J= 
DU 
D^U-^DH.D^U 
3(m“2) 
(P. 
□ a ; 
(m-l) (to-T)H 
a formula which will be presently useful. 
56. The foregoing equation may be written 
(D^u)— 3( _ DD^U + |DTJ) 
+ (D^U)-^DU{(m-2)n^D^U-f(m-3)nD=>U-iD^U}+&c.(DU)=' 
and the term — lID'U+fD^U is equal to 
1 
.. = 0 ; 
|(D^U- 
H 
DH . D^U - ADU . D"“U = f JD U - ADU . D^'U. 
Substituting this value the equation divides by DU, and throwing out this factor it 
becomes 
(D^U)“-\|J-AD^U) 
+ (D^U)“-n(m-2)rFD^U-|(w-3)nD=^U-iD^U}+&c.DU=0; 
oi obserAong that fnD^U=n^D^U-i- term containing DU, this may be written 
(D^u)— 3(|j_AD^U)+(D^U)“-Xn^D^U-iD^U)+&c.DU=0. 
57. Ii the equation dhides by DU we shall have a five-pointic contact ; the condition 
for this is that 
- A(D^U)=*+|JD^U + n^D^U -iD^U 
may divide by DU, or more simply that 
- A(D^U)^+|JD^U-}-| (^^DHyD^U-iD^U 
may divide by DU, or, what is the same thing, the function in question must vanish in 
virtue of the substitution of the values Cx~Av, A;u,— BX in the place of X, Y, Z. 
3 G 2 
