116 



Sir J. W. Lubbock o?i some Methods of developing 

 2nd Operation. Elimination of B 3 and A 3 . 



(2)i + (2) 3 -2(l) 3 cos3 = (3) 1 

 (2) 2 +(2) 4 -2(2) 4 cos3 = (3) 2 

 (2) 3 +(2) 5 -2(2) 5 cos3 = (3) 3 



Taking any vertical column as that under B 9 , the first operation 

 by which A x and B x are eliminated, is — 



(l)i + (l) 3 -2(l) 8 cosl = 2 # 9 {cos9 + cos27-2cosl8cosl} 



+ 2^4 9 {sin9 + sin27 — 2 sin 18 cos 1} 



= 2 Sgsin 18 {cos 9 — cos 1} 



— 2 ^4 9 cos 18 {cos 9— cos l}. 

 If, as before, 



2 sin 1 sin 1 be represented by . . . . i 



2 2 sin 1 sin 2 be represented by ... 2 



2 3 sin 1 sin 2 sin 3 be represented by 3, 

 &c. 



The following table exhibits the entire factor. 

 Table VIII. 



Expressions may also be obtained by means of which A t and B t 

 can be obtained at once whatever i may be, without depending upon 

 any others with different indices. 



