314 Proceedings of the Royal Society 
assume (since but three distinct and non-coplanar values of 8 are 
required) 
8 = x y 
where y is a constant unit-vector, and x a scalar function of t. 
This gives 
x ~ 
- 1 = 9*! • 
x 
The values of y are therefore unit- vectors parallel to the axes of 
the surface 
S p(pp = 1 , 
and those of - are the roots of the auxiliary cubic in <p . Call 
x 
them rj 1 , y. 2 , y A and g iy g, ly g s respectively, then the values of 8 (into 
which no arbitrary constant need be introduced), are of the form 
jt 
g y. 
Thus, finally, 
p = — %y^yp 
= - [fs^Syadt + C] . 
5. If, in the equation of (4), we suppose a constant, we may 
easily apply a process similar to that of (2). 
For 
p = p + pSt = (1 — St . <Q) p + a St . 
Hence, as a is constant, 
/ v ( i ~ -T - 1 
-T I 1 _ ) , x V n J . 
P n )Po + ■^ 00 ^ 1 _ _ 1 n 
= *~ t<P Po + @ a 
where p 0 (which is arbitrary) has been increased by <p -1 a. It is 
easy to show r that this agrees with the final result of (4), and the 
coincidence is so far a justification of the use of the method of 
separation of symbols. 
The verification of the general result of (4), where a is variable, 
can also be effected by this method, but not so readily. 
6. Let us take the linear equation of the second order with 
