294 
MR. LUBBOCK’S RESEARCHES 
$ A, = A,, sin {nt — n t t ) + A (2 sin (2 n t — 2 n t t) + A /3 sin (3 n t — 3 n,t) + e A, 12 sin (n t — 2 + w) 
+ e A, 13 sin (2 n t — 3 n,t + ro) + &c. 
Supposing that the arguments 1, 2, 3, 12, 13, 31, 32, 64, 65, 73, 74, 
112, 113, 155, 174, 213, 273, and 312 are alone sensible in S . S X, 
§ and 5 X, the coefficient of e 3 cos (2 nt — 5 nf, + 3 vr) in the expression for 
** or ii* 155 
+ T { 3 R “ - 7 M {*• - M - T (t?* + t} ■"> 
+ ^152 ~ 4 R 158 j> ^ A 3 — A /3 ^ 2 j a ~ r 'w 4" ~~2 { ^ 12 — j” 
+ R , 2 { A , 3 - \,s } - r 'u + 2 R 6i { A n - A,„ } 
f y y 1 _ fl ^ »' P fy -if 
< A 10 — A ;I0 > 2 (| a ~ r 73 •“'198 1 a 73 a /73 J — 
/ a i 4 • Rl54 i a i 4 • R 1 56 \ _ I 
2 1 d a, d a l J 1 1 
a d • R 5 2 .. 
~2d^ 13 
a d . R 65 f 5 p 
~TdcT 10 2" 
a d R lp3 , 
2 d a 74 
1 _ ad - fi o r r 
2* R^93 I ^74 — ^/74 J - r ' 
1 f a, d . i? I53 . a ; d . Hu, 4 > _ 1 /^dfi^o x ^d J? 158 \ ^ \ a ( d . R b3 ^ , 
'2" l da, + da, J ' 2 "2 l^d^” + “d^J ' 3 2d _ a” M 
a , d . 
2 da. 
a, d . U(|4 r 1 
2 da. 
/ n 
a ( d.R 65 ^ ( a ; d R ls2 a 
— ^ — t ' i in ~ — ; ' /7<t ■“ 
2 da 
; d i?i 03 ^ a < d . R 0 ~ 1 
2 da, ' /73 2d a, ,74 da, M 
In the same way the expression for & . R m , & . i? 213 , S . R 273 , and . j? 312 may 
be found from the preceding Table. 
If a < a, and 
1 1 — COS 0 + ^| U = ib l>0 * + 6 1(1 C0S 0 + 6 1)2 cos 2 0 &c. 
{ 1 — — cos 0 + 1 “ = \ b 3 o* + 63 ! cos 0 + b 3 2 cos 2 0 &c. 
L a, a/ J J 
^ = W/ ( C ° S2 2" “ C ~ ~2 C ' g } C ° S ( w 1 ~ n < 0 t 
3 m, a , m. a , n . . , , 2m,a , , _ 
5-' — e cos (n / < — to - ) -f- — i— e cos (2 nt — n,t — -a) + — J— e , cos (n t — 2 n, t + ra - .) 
4 O. fl/ f n 2 
2 a 2 
* The notation is slightly changed from that used before, 
t e and e ( which accompany n t and n , t are omitted for convenience. 
