1880.] some theorems in trigonometry. 385 



§ 15. If (a)', (b'Y, (c'y, (d'y denote the quantities formed from 

 a , h\ c, d' by the same process as a , 6', c, d' were formed from 

 a, h, c, d, and if a corresponding meaning be assigned to 

 (a')", ... {a"y, ... (a'y, ... then it can be shown that we have the 

 four sets of equations 



/ '\/ / tt\// f ri\i r I t\tr // 



\a) =a, [a ) ^a, [a ) = — a , {a) = a , 



■ {by=b, {h"y' = h, {hy= y, {hr=-h'\ 



c) =c, (c ) =c, (c ) = c, (c) =-C , 



{d'y=d, {d"y'=d, {d"y= d\ {d'y'=-d". 



§ 16. Taking the known formulge 



8 cos a cos h cos c cos d = cos 2a' + cos 26' + cos 2c' + cos 2d' 

 + cos 2a" + cos 2b" + cos 2c" + cos 2d", 



8 sin a sin b sin c sin c? = — cos 2a' — cos 26' — cos 2c — cos 2d' 

 + cos 2a" + cos 26" + cos 2c" + cos 2d", 

 which may be written 



8 n (cos a) = S cos 2a' + S cos 2a" (i), 



8 n (sin a) = — S cos 2a' + 2 cos 2a" (ii), 



we obtain from them by subtraction and addition the formulae 



4 {n (cos a) - n (sin a)} = S cos 2a', 

 4 [II (cos a) + n (sin a)} = ^ cos 2a", 

 the former of which is [E), (§ 7, p. 823). 



Substituting a, h', ... and a", 6", . . . respectively in place of a, 

 h, ... in (i) and (ii) we have in virtue of the equations in § 15, 



811 (cos a) = S cos 2a + ^ cos 2a" (iii), 



Sn (sin a') = — 2 cos 2a + % cos 2a" (iv), 



811 (cos a")= S cos 2a' + % cos 2a (v), 



811 (sin a") = — 2 cos 2a' + 2 cos 2a (vi), 



and writing for the moment (2a'), (2a"), ... in place of 2 cos 2a, 

 2 cos 2a', . . . the formulae (i), . . . (vi) are 



sn (cos a) = (2a') + (2a"), 8n (sin a) = - (2a') + (2a"), 

 8n (cos a) = (2a") + (2a), SH (sin a) = ■ (2a") - (2a), 

 8n(cosa") = (2a) +(2a'), 8n(sina")= (2a) -(2a'), 



from which we deduce that 



n (sin a) = n (sin a) + IT (sin a"), 

 n (cos a) = n (cos a') — II (sin a"), 



28 2 



