544 
Proceedings of the Royal Society 
where the letter f is to stand for either + 1 or - 1 as the case may 
afford. This gives by (36 bis) 
n — ~Y (gbh - b 2 az) 
= i-[(. h-V*)az-hYf]. 
Introducing the value y into Y (38) we get 
Y/= (h-b*) 
As we need only the terms of the first order in ( h - b 2 ) we may re- 
place hY by b 2 Y. Thus 
n — \ (h - b 2 ) [ az - b 2 Jl - w 2 ] 
The coefficient of h in (37) becomes 
z(h - M 2 ) + a . bg — \Ji - M 2 + a 2 ]z - ( h - b 2 )a Jl - w 2 ; 
and considering the value of M 2 the coefficient of z will be (h — b 2 ) . 
Also in the term jy(h - a 2 ) we replace y by 
(h ~b 2 ) — . 
(y 
The term becomes, for h — a 2 — b 2 - a 2 = - c 2 , 
(h - b 2 ) — (b 2 - a 2 ) = - (h - b 2 )wc . . 
.0 
We have now the factor ( h-b 2 ) existing in evidence in all the 
terms of the relation (37), which becomes rigorously as to the first 
order in ( h - b 2 ) , 
0 -{h- b' 2 ) 
I 
i 
W jjH - b Jl - 
+ ix - jew + k(z - a Jl - w 2 ) 
As the second member represents the terms of the first order in 
(Ji - b 2 ) , and as h - b 2 is variable, we are entitled to equate the 
factor of h - b 2 separately to zero. 
