210 Proceedings of the Koyal Society of Edinburgh. [Sess. 
With the values of /3 and V that have been obtained, we now have 
/? r 
iV. 
2 8/ ^ 
and so the equation 
becomes 
Hence we have 
Pl _l ^ ^ _l Pi ^ ^ 
^ V /3 V S “/12 
1^11^ _ ^ ^ ' 
^8'2 8 
n^=l 
and without loss of generality we can take 
n=l . 
Finally, the equation 
being 
becomes 
M = 0, 
r /3 
which is satisfied by the values of /5 and T. 
Hence all that remains to be done is to integrate 
^11 ^ _ 4^1 — 4p_ = 0 . 
8/2 
Lm 
For the purpose, take a 
new variable u such that 
8j^' = ?^82 ; 
then 
and therefore 
^11 0^1 _ 
8/ 8 t/8 ’ 
The equation becomes 
and therefore 
Um\h 8/’ 
where h is a constant of integration. Hence 
