314 
Proceedings of the Royal Society 
assume (since but three distinct and non-coplanar values of 8 are 
required) 
8 = x rj 
where ^ is a constant unit-vector, and x a scalar function of t. 
This gives 
x 
- 7] = <pr) . 
X 
The values of g are therefore unit-vectors parallel to the axes of 
the surface 
S p?p 1, . 
and those of - are the roots of the auxiliary cubic in <p . Call 
x 
them y]^ rand g n g 2 , g. 6 respectively, then the values of 8 (into 
which no arbitrary constant need be introduced), are of the form 
s gt v . 
Thus, finally, 
p — — r]p 
— — 2s ~ Vf J s^Srjadt + Cj . 
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 + p8t = (1 — 8t . <p) p + a8t . 
Hence, as a is constant, 
= £ t4> p 0 + <P « 
where p 0 (which is arbitrary) has been increased by tp' 1 a . It is 
easy to show 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 
