CONTACT AT ANT POINT OF A PLANE CUEVE. 
370 
BiU, &c., the symbols B^ contained in Bi and Bg operate only on U, and 
not on the variable quantities A, B, C, &c. contained in Bj and Bg. 
18. If now we treat Bj as an operand, that is, perform the differentiations on the 
variable quantities A, B, C which enter into Bj, we obtain 
Bj.Bj^B^, 
or, Avhat is the same thing, operating on BjU with Bi, the result is 
B,.B,U=B?U+B,U=:(B?4-B,)U, 
and in like manner 
B,.B*U=:(Bf+2BA)U, 
B..B?U=(Bt+3B?B,)U. 
It is, in fact, upon these principles that Hesse’s values of BjU, BfU &c. were obtained, 
and we may by means of them obtain the other expressions assumed in the preceding 
section. 
19. In fact, starting from Hesse’s equation, 
B^U=-^OU-7-^HB^ 
'■ m—l (m — 1)^ ’ 
we have 
(B;+23.3,)U=;^j (UD<I>+<l>DU)-;;;Ap(DH.S-+H.2&Da). 
But we have identically DU=0, DB=:0, and this equation becomes therefore 
(3t+23.3,)U=^UD<t-5Ai)iDH.a=. 
But this is precisely Hesse’s value of B®U, or we have B,B2U=-0, and therefore 
(d;+33.3.)U=3;U=^UD'I>-(;AFDH.a“. 
20. In like manner, starting from the expression of BiU, we have 
fit 1 
(Bt+3B?B,)U=^(DU.D<I>+UD.D<1>)-(;;^.(DH.2^])^+^'^D.DH); 
or since DU and D^ vanish identically, and the values of D-D^, D-DH are B^tP-hD'-^'- 
b,h+d^h, we have 
(3i+ Sdjaju =~ u(S,<i>+d’3«)-(Aip a^ca^H+D’H); 
and if from the double of this equation we subtract Hesse’s equation, 
a:u= A tu(d-o- At n)-,ATpy(D=H-^+M h<i>), 
we find the required relation, 
0:+63;5.)U=Ai(23,'I>+D’'I>+At''J) U 
-(At?(23.h+D’h+ Ai- tTf H<p)a-. 
