23 
4] IN SERIES OF MUPTIPLE SINES AND COSINES, 
(43). 
substituting in 
cos k{0- 0') = cos k (tz- tz') {cos k (0 - tz) cos Jc (0' - &') + sin k (0 - tz) sin k(0' - tz')} 
- sin k (tz - tz') {sin k (6 - tz) cos k (0' - tz') - sin k (0' - tz') cos k (0 - tz)}, 
and reducing the whole to multiple cosines, the final result takes the very simple form 
cos k (0 - 0') = cos [r'm' -rm + k(iz- tz')} A r AV cos kL cos kL' (l - (u - 7 V) .. .(38). 
kr) \ kr‘ 
Again, formula3 analogous to (14), (18), may be deduced from the equation 
r (m 1( m 2 ...) 
^ ^ x H" dm x f 2 " dm 2 
'-«2-w* cos(r 1 m 1 +r 2 m 2 ...)J J ... c°s (r^ + r 2 m 2 ...) T (m 1} m 2 ...) 
+ sin(nwij + r.,?n 2 ...) J ^ J ... sin (nmj + r 2 m 2 ...) F (m 1} m 2 ...) (39), 
which holds from w x = 0 to = 2tt, &c., but in many cases universally. In this case, 
writing for T (m u ra 2 ...) the function 
n 
\/l — e 2 — e sin u J — 1 
/{ e «iV<—i>, 6 « 2 V(-d ...j ..,(40). 
- 1 e wV 1-11 {e wV <~i) j 1 — e cos u 
and observing 
Jl — e 2 — e sin u J - 1 1 + Xe -M * r(-1) , . ,— . 
1 — e cos w = 1 -x e -"V[-.i = 1 + 2S , ( cos l an ««] V (41), 
an exactly similar analysis, (except that in the expansion 
r^, x,..)=2_:2_:...a Si>S2 ...va/ i ..., 
the supposition is not made that A Sii Si ... = -4_ 8li _ Sa ...), leads to the result 
J 1 — e 2 — e sin u J — 1 ) 
y{f M i^< g w »V(—i),, , j pj 
J — 1 e MV(_1) % / {fc MV(_1) } (1 — e cosw)) 
^ 00 00 
** - 00 “ - 00 • • • 
cos (rA + r. 2 m 2 ...) 2 n cos (r^Xj + r 2 y 2 X. 2 ...)/(V, X 2 ...) 
+ sin (rpiij + r 2 w 2 ...) 2 n sin + r 2 % 2 X 2 ...) /(Xj, X 2 ...)... (42), 
(n) being the number of variables u 1} u 2 .... Hence also/{e MlV(-1) , e M * V( '" 1) ...} 
-- cos (« + ...) cos (r^X, + ...) n ■))| /(Xl , X,...) 
'C* 00^ 00 
- 00^— 00 •* 
i +sin (r,m, + ...) sin (*»X. + ...) n ))[ /(Xl , X,,.)