590 
Proceedings of the Royal Society 
cuius somewhat more intelligible to the beginner than the methods 
employed by O’Brien and Murphy, whose works on the subject are 
usually read in this country. As I am not writing a treatise, but 
merely sketching a method, I shall run over the principal elemen- 
tary propositions only. 
1. Let 
1 1 
P 
\i — #Q i 
'(i-2 hfx+h 2 y> 
This is possible, if h be always taken less than 1 ; and, as //, is never 
beyond the limits db 1, 1, Q*, - 1 are in order of magnitude, and 
the series is always convergent. 
Hence we may differentiate, and we thus obtain 
d L 1 
d/x p 
p 3 " ^ 
and 
l( a -'A) - w+3{i -,■)»• } 
Also 
and 
72 d 1 uW-li 3 . 
h tt ~ = 3 — = S Qi , 
dk p p 6 ’ 
dh p) ^ 5 { — P 2 + 3 (/x — A) 2 h 2 1 
= 5 . a (V 4- 1) A ? ’Q* 
( 1 ). 
( 2 ). 
The sum of the multipliers of p~ 5 in (1) and (2; is obviously zero. 
Thus we have the equation for Qi 
i(i+l)Qi + ((1-P*)f) =0 . (3). 
2. From this equation, by differentiation s - 1 times with respect 
to fi, we have 
