374 
MK. A. CAYLEY ON THE CONIC OF FIYE-POINTIC 
and reducing as before, the consecutive value of D^U is 
(m-2)b!U 
8. Substituting in the equation D‘U-n.DU=0, we obtain as the conditions of a 
live-pointic contact _ (^_ 2 p 5 u+p,i 3 ;u= 0 , 
_i[(m-l)3;+ 3(m- 2)3,3JU 
+ P.i(3!+33,a,)U+3V--i3!U=0, 
_^[(m_l)(3t+63;3,)+(m-2)(43,33+33?)]U 
^_p,^p;+63;3,+43,3,+33aU+3,P-i[3'+33|3JU+i3=P-iS;h=0; 
or reducing 
P=2(m--2), 
s P-i 
O.t^— 3 
r 3 ; + 63;3ju 4 3;o [ 3 ; + 33,3JU^ 
3.p=i — 5ju 9 3;u 3V 
which are the conditions of a five-pointic contact: it is to be remarked that if’ only the 
first and second conditions are satisfied, we have a four-pointic contact, and it onh 
first condition is satisfied, a three-pointic contact. n , . n r ,ve the 
9. We have to reduce the last-mentioned equations; suppose that A, B, h aie 
first derived functions of U, then the equation 3.U=0 may be written 
Kdx + ’ 
and this will be satisfied identically if 
— Cf//, 
dy=.C^ — Av, 
dz =A/u-— B?t, 
where X, ^ are arbitrary multipliers, which may be taken to be constants. AN e luuc 
therefore bi = D, where 
]} = ( Bn ~ + (CX - An)B^ + ( Ap - Bx)b,. 
10 . The resulting expressions for 3;U. SfU, 3;U may be exhibited in the reduced 
forms given by Hesse, viz. if ^^^Kx+y^y+vz, we have 
b‘fU = P3U-Q2A\ 
B?U = P3U-Q3A% 
hfU = l^U-Q,A^ 
where the values of V„ P„ P. ; Q, Qi, Q. aie as follows, viz. if («, 5, c,f, J. *) are the 
second derived functions of U, and if 
