24 ON CERTAIN EXPANSIONS, IN SERIES OF MULTIPLE SINES AND COSINES. [4 
By choosing for e u ^ { ~ 1) functions expansible without sines, or without 
cosines, a variety of formulae may be obtained: we may instance 
(A, - A- 1 ) {1 - je (\ + q 
Jl - e 2 - \e (A - A" 1 ) 
(44), 
A r having the same meaning as before. 
Also, , ( « X 
’ Jl — e 2 — ^e (X — A -1 ) v ' 
N y 
where Mr = A’6~? ( "“ X ' 1) - A- e? ^ (46). 
Again, ji_i e (A + A- 1 )} (A - A" 1 ) M r + 2 ^ Q (47), 
and jl-I^A + A *)} {(A + A *) A >- = (ITi e ^ + x -i)[ {(A + A" 1 ) - \e) A r , ...(48); 
(A - A" 1 ) " / 
i-i 
n/1- 
or, what is the same thing, 
(A - A" 1 ) 11 - \e (A + A" 1 )} {(A + A" 1 ) - ¡¡j~A r Q 
Vl -e 2 
(49) ; 
or, comparing with (44), 
(50), 
which are all obtained by applying the formula (43) to the expansion of ^ (0 - -sr), 
and comparing with the equations (25), (33).