G04 
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 r 
Thus 
i. a d R i i ■ d 
R ‘ = - TJT + ,R ‘ 
D adR. , . D 3adi2, 5iR, 
2R '° = - JJ7 +,R ‘~ -4d„ + ~ 
_ 1 r_a a d 2 72,_ad7?,,^ a di2 1 'l 
2 1 2d a 2 2da da J 
iadZ?, „ 3adZZ, , 5iR t 
~2dT + K ' ” ~4dT + ~T~ 
p _ a 2 d 2 R, _ (2 i + 1) a d Ry _j_ (4i® + 5 i) Z? ( 
K ‘° ~ Tda®~ 4 da 8 
Changing the sign of i, we get 
p _a®d 2 Z£ 1J (2i— 1) adR, (4i- — 5i)R l 
Q ” 8 da* + 4 “da - + 8 
which accords with the expression (for N < '° ) ) given in the Theor. Anal. vol. i. 
p. 463. 
„ „ adZt 10 , • p 3 adii t .5 ip 17adiZ 1 13iZZ, 
i Rii = ~ ~da~ + lR '°~~4 "da - + T “ 16 “da" + 8 
J_ f a* d 3 4 ZZ, a 2 d 2 R, _ (2 » + 1) a 2 d® R, _ (2 i + 1) adfl, (4 i + 5) idfl, "I 
— 2 1 8da s 4da 2 4 da® 4 da 8 da J 
. f a 2 d R, (2 i + 1 ) a d R , (4 i + 5) i ZZ, f 
l 8 da* 4 da 8 J 
3 | a-d 2 Ry _ adR x ia d ft, 1 
4 1 2d a* 2da da J 
, 5i f 
+ T\ 
a d /2 , 
2 d a 
+ * 
M 
17 a d 72 , 13 ^ 
16 da + 8" 
R„ - 4 I (26 i + 30 i* + 8 t s ) ZZ, - (9 + 27 i + 12 j*) 
48 L da 
a 2 d 2 ZZ, 
+ (6. + 6)-j- s -! 
a 1 d’ZZ, 1 
da 3 J 
Changing the sign of i, we get 
