494 ON A NEW ANALYTICAL REPRESENTATION OF CURVES IN SPACE. [294 
The complete value must, it is clear, be of the form 
a 2 (b 2 + c 2 4- d 2 ) V + k (ps + qt + ru), 
vanishing in virtue of the equations V—0, ps + qt + ru = 0, and this being so, observing 
that V contains no term in ps, we have k = coefficient ps in 
(a 2 4- b 2 , c- + a 2 , a 2 4- d 2 , — cd, + bd, — cb) ( U, T, P) 2 , 
that is 
k = 2 (« 2 + 6 2 , c 2 + « 2 , a 2 + dr, — cd, + bd, — cb) (— Z>cZ, cd, № + c 2 ) (cu, ha, 0), 
or 
= (a 2 + Z) 2 ). — bd .ca — cd {cd . 0 + (b 2 + c 2 ) ba} 
+ (c 2 + a 2 ). cd . ba + bd {(6 2 + c 2 ) ca — bd. 0} 
+ {or + d 2 ). 0 — c6 {— bd .ba -\- cd. ca], 
which is 
= abccl I — (a 2 + b 2 ) — (b 2 + c 2 ) 1 , =0. 
-j + (c 2 + a 2 ) + (b 2 + c 2 ) 
V + (b- - c 2 ) ; 
The coefficient k consequently vanishes, and therefore in T) 2 F the coefficient of ^4 2 a 2 
is a 2 (b 2 + c 2 + d 2 ) V, but I have not worked out the coefficients of the other terms. 
2, Stone Buildings, W.C., SOP October, 1860.