382 
PROFESSOR G. H. DARWIN ON FIGURES OF 
the last transformation being derived from (7) with i -f- 2 in place of i, and k -f- 1 
in place of k. 
Differentiate (10) with respect to p, and notice that dr*/dp = 4, and we have 
r^-2(2i-l = 
dp 1 . 'dp dp- 
Then, with i -f- 2 in place of i, 
i d^Wi y g 
Now 
2 (2 i ' + 3 ) ^=^ 
T P (A = ~ 2 < 2i ' + 0 w < 
1 dw. 
t + 2 
y.2i + 3 
Differentiating again, 
dp 
by (10). 
d- wi 
dp 
2 ™2i + 1 /j*2t + 5 
1 f o (1“W{ + ^ o /o ’ l o\ dVy'i + 0 
r > rv-2 (2i +3) 
dp 3 
dp 
or 
1 d 2 
= y^T+5^2^+4 by (11), 
1 d 2 W; + 2 d} Wi_ 2 
>«2i + 1 
dp 2 dp 2 T 
,2 ,.2i-3 
(n) 
( 12 ) 
But since p = i(« 3 + y 3 ), it follows that, in operating on a function involving x and y 
only in the form cc 3 + /> 
d 
dx 
d 
d 
d 
d 
Also 
so that 
-■'"V and 
* = l± + i~*£. A = i 
da; 2 2 dp ^ 4 ' dp 2 ’ dv/ 2 2 dp 47 rfp 2 
Now let us put 
Then 
d 2 d 2 _ j , 2 2X d 2 
da: 2 d^/ 2 — 4 ^ dp 2 
* 3 _ d^ _ dl ^ 
6 da: 2 d f ’ 
and therefore (12) may be written 
(13) 
Wi. 
