604 
MR. LUBBOCK’S RESEARCHES 
obvious that they may be reduced so as to contain only the quantity R x and 
the differential coefficients of this quantity with respect to a and a . 
Thus 
R* 
adE, 
2d a 
+ iRi 
2 R w — 
adR 4 ■ „ 3adR t 5iR l 
2d^ +lR *--4d a + ~T 
_ L I — a-d'-R i _ R , , i a d_K_i 1 
2\ 2d a 2 2da da J 
iadi? t . 9 P 3 a d R t , b iR l 
2da 1 4 d a 4 
„ _ a 2 d 2 R { _ (2 i + 1) a d R l (4 £ 2 + 5 i) R l 
10 — 8 d a 2 4 da 8 
Changing the sign of i, we get 
p _a 2 d 2 R 1J (2i — 1) ad R, (4i 2 — 5i) R y 
9 ~ “8d^ + 4 ~d~cT + 8 
which accords with the expression (for N^) given in the Theor. Anal. vol. i. 
p. 463. 
, } t> _ ad R\o ,-p 3 adE, 5i „ 17 a d iq 13 i R, 
3 R i2 — — + lR io — -^ +T^ _ l6"dr' + ~8~ 
1 f a- d 3 R t a 2 d 2 R, _ (2 i + 1) a a d 8 K 1 _ (2 i + l)adR 1 (4i+5)idE 1 'l 
2 1 8da s 4da 2 4 da 2 4 da 8 da J 
. Ja 2 d R t __ (2i + 1) adE[ (4i + 5)ii? 1 'l 
1 8 d a 2 4 da 8 J 
a 2 d 2 R! _ adE! x iadR 
" 2 d a* 
2 d a d 
LM 
a / 
, 5i f adfij , . 
+ T\~^dLT +l 
L-'L-Lr + 'i*' 
Ru 
-L ( (26 i + 30 i 2 + 8 i 3 ) R, - (9 + 27 i + 12 i 2 ) 
4o L da 
+ (6i+ 6) 
a 2 d 2 J?j 
da 2 
a 3 d 3 R t | 
da 3 J 
Changing the sign of i, we get 
