532 Proceedings of the Royal Society 
Owing to (6) the coefficients of the cubic may he expressed hy 
' m =0, 
(13) \ m i~ ~ j 
; m 2 = - + Sj'<f>j ) . 
We have now from (7) 
h' — (<£' + w)p _1 ; 
and putting 
(14) m u = + mpt + m 2 u 2 + , 
we get 
m u h' = [???<£' _ 1 + u(m 2 - <£') + it 2 ]p . 
The term mcf>~ 1 p being indeterminate, owing to m — 0, we 
consider 
m<h' ~ ■‘VAju, = \ cfiXg>p, ' 
and putting VXp, = p , we get 
(m^-V)= - V^/S&'p 
(See also Proc. P. S. P., Dec. 1880, on the case m — 0) . 
We remark that hy (7) we have 
Ship — — u . 
Substituting this in the preceding term we get 
(15) — “£' = Y + (ni 2 + u)p- 'p. 
'll/ 
Effecting the combination 
K + </>>) - (tw + K + «)p)2 = 0 , 
and remarking S k'<j>p — S /)</>&' = 0 , we get 
o = ($ P y - Y - vvv/ - (™ 3 + «) v 
- 2(m 2 + u)S p4>i'<j>j' • 
( 16 ) 
m, 
We obtain a relation between — and (m 2 + u) independent of m v 
u 
Namety, we introduce into Skf = 0 the value 
7 =v(;>;' +/</»/) . 
