596 
Proceedings of the Royal Society 
where Q* is as in § 1, and P* is a function of /x and <p. The 
equation (10) becomes successively 
d_ 
d/x 
/ ox d CPiQi)\ 1 d^PiQi) _ 
(a-^) (/; , ) + rr^ dip 2 + <*+i)P.Qi=o, 
Q t d 2 ?j 
— jj? dtp 2 
jp a, j-p, 
and, finally, 
+ =°- 
If we put, for a moment, 
= 0, 
djx 
(i -*■)<** 
(which has a real meaning, see § 11), 
and suppose Q* to be expressed in terms of v instead of /x, calling 
it q i3 the equation may be written 
d 2 Vj 
dv- + 
( 24 ). 
Hence it appears at once that P i cannot contain <p except in the 
form of factors, such as cos. s<p, sin. stp, in the several terms of 
which (as an integral of a linear equation) it must be composed. 
Hence, as before, 
to 
P,; — h-s ®i COS.(sp + a), 
and, keeping to one value of s, 
