TO THE THEORY OF EVOLUTION. 
9 
and this into 9'°j |5 is generally quite insensible. Very often two or three terms on the 
right-hand side of (xxiv.) give quite close enough values of 9, and accordingly of 
r = sin 9. (xxiv.) is clearly somewhat more convergent than (xix.) if Ji and Jc are, as 
usually happens, less than unity. 
Returning now to (xix.), let us write it 
e h, Jc). 
This is the equation that must be solved for r. Suppose r 0 a root of this when we 
retain only few terms on the right, say a root of the quadratic 
e = r -f- Tjhhr 2 . 
Then if r = r 0 + p, 
e =/( r o> K fy + pf'{r 0 , h, Jc) + \jpV"(r Q , h, Jc) + &c. 
Hence p = 6 to a third approximation 
r f(rjik) 
x/l-ro 2 
iQ* - 1) (V - IK 3 , 
1 Inearly . 
e n-v 
(xxv.), 
which gives us a value of p which, substituted in p 2 in the above equation, introduces 
only terms of the 6 th order in r 0 . 
Another integral expression for e of Equation (xxi.) may here be noticed : 
_ e W+V)[ r . - d L - g-5, 
£ {h 2 +i 2 —2rhlc) 
Put h = -n 06 + y), Jc = — (£ - y). 
Hence 
J o v t — r 
— gK^+V 2 ) 
f 
J 0 
dr 
5 ‘V l + r 1 l—r)' 
1 — r 
Let tan 2 d> = --, or, r = cos 2 <A 
T 1 + r 1 
Therefore 
f45° 
e = 2 e i032+v2) g-ltf 2 tan>* + 7 * cot? 4.) rft 
u 
2e 
e -i(S + yV ) 
1 + v 2 
where v = cot <£ and is > 1 . 
It seems possible that interesting developments for e might be deduced from this 
integral expression. 
vol. cxcv. —a. c 
