122 Sir J. W. Lubbock on some Methods of developing 



So that, finally, for example, writing the indices of sines outside 

 the bracket if B }3 and A l3 are neglected together with every B and 

 A which haB a higher index than 13, 



(6), = 2 10 2.1.3.2.4.3.5.4.6.5 { B x cos 6 + A x sin 6} 

 (6),= 2 10 2.1.3.2.4.3.5.4.6.5 {B x cos 7 + A x sin 7} 

 (6) 1 sin7-(6) 2 8in6 = 2 10 .2.1.3.2.4.3.5.4.6.5.1.#i 

 (6) a cos 6 - (6) sin 6 = 2 10 .2.1 .3.2.4.3.5.4.6.5.1.^. 



If the A\ are missing, then if jB 7 is neglected with every B which 

 has a higher index than 7 j 



(3) 1 = 2 4 .3.2.4.1cosl5J5 5 . 

 If B 9 is neglected with every B which has a higher index than 9 ; 



(4) 1 = 2 e .3.2.4.1.6.1 cos 20 B 6 . 

 If B n is neglected with every B which has a higher index than 



(5), = 2 8 .3.2.4.1.6.1.7.2cos 25 B 5 . 



The series may be separated into four categories by using the 

 particular values of the series due to the angle 1, —1, 180° + 1, 

 180° -1, 180° + 2, 180° -2, &c. 



The 1st category contains ZPs with even indices. 



2nd ..... B's with uneven indices. 



3rd A's with even indices. 



4th A's with uneven indices. 



The expressions which are required in this case may be inferred 

 from those which have already been given. 

 Let 



i?, j denote the value of the series R due to angle i, 



n lt3 i8o°+ 1, 



i? 1>4 -180°-1, 



fi ]{ =zB + /? > iCosl+J5 2 cos2 + &c. + ^ 1 sin l+^ 2 sin2 + &c. 

 2*1,2= .Bo + B i cos 1 + ^ 2 cos 2 + &c— -4, sin 1 —A^ sin 2 + &c. 

 B l3 c=B —B l cos 1 + 5 2 cos 2 + &c. — A l sin 1 +A q sin 2 + &c. 

 B i4 =B —B 1 cos 1+ 2*2 cos 2 + &c. + A x sin 1 — A q sin 2 + &c. 



