On some Methods of developing any Function of Sines, S?c. 107 



Le Verrier has given expressions by means of which A { , B { are 

 determined, the A's and B's having lower indices, being obtained 

 successively from these, by means of the particular values which the 

 series assumes when the arbitrary angle which I have taken equal 

 to unity becomes successively 1, 2, 3, 4, &c. 



The following table shows the order in which the indices recur. 



Table I. 



1 st Operation. Elimination of B . 



(I) 1 = j? -# 1 = i? 1 (l-cosl)+5 2 (l-cos2)+£ 3 (l-cos3) 



— A x sin 1 — A 2 sin 2 — A 3 sin 3-|-&c. 

 (l) 3 =i? 1 -/? 2 =2? 1 (cos 1 -cos 2)+Z? 2 (cos 2-cos 4)+Z? 3 (cos 3-cos 6) 



-\-A x (sin 1— sin2)+^ 2 (sin2— sin4)+^ 3 (sin 3— sin 6)+&c. 

 (1) 3 = 7? 3 — 7?3 = -B x (cos 2-cos 3)+l? 2 (cos 4 -cos 6)+2? 3 (cos 6-cos 9) 



■j-Ai (sin 2— sin 3)+^ 2 (sin 4— sin 6)-f-.^ 3 (sin 6— sin 9)-)-&c. 



The following table shows the order in which the indices recur. 

 Table II.— Indices of variable factor. 



which table shows that in (1) 3 , for example, 



