and Cosiries of Multiples of a variable angle. 265 



But wheu^=/, the coefficient of A,- is 



sin ink 



(, , X , \ SH 



2a-\-[n — \)k I— r 

 ^ ' / SI 



sin ik ' 



Suppose Am the last coefficient which is sensible, and let p , 

 have all the values 1, 2, 3, .... m except i. As the coefficients 

 of Ap and Aj differ from those found in (4.) only in some of 



their signs, the same conclusions result from them when we 



2_ 

 make k= — , and k=z2a. Therefore, passing by the formulse 



^ 27r 



derived from k= — , as we have before done, and taking only 



that which results from — =«= — -, we have 



2 2n 



A,= I X^{{2i'+1) ^} sin i2i'+ 1) ^, n > 7«, . (9.) 



where i' has the same values as before. 

 This may be reduced as (6.) was. For 



• /^ , \ ^i" — • 2V . ,^ ^v zV _ . S/tt « 

 sm(2« — 1)— =+ sm— , sm(2n—3)——+ sm— — ,&c. 

 ^ 2w 2n ^ 2n 2n 



If therefore we make Wj, Wg, &c. the sums of the first and last, 

 of the second and last but one, &c. of the particular values of 

 '^{x)i and ^1, /g, &c. the differences of the same, (9.) may be 

 replaced by 



. 2/ . i-Jt , . Silt , , . ln—\)i'n 



^'■= ;rr''"^2^+^^'^"-2;r +--+^i'^"-W- 



or 



. 2 /, . i-TT , ^ . S/tt ^ , (n—l)iTr 



^■■= n V' ''" 2S +'» ™ ^ + •••• +% "" -1^ 



the first when i is odd, the second when it is even. 

 We now proceed to the more general form, 



(10.) 



yi[a:) = Bo+B, cosiT + BgCOs 2x+ ,.., 

 + A, sin j; + Ag sin 2a? + 



;;;;} . . (c) 



The solution of this form might be derived from those of 



the two forms before treated, by taking that particular case in 



27r 

 which k= — ; but I prefer treating it separately. 



y(a) = Bo+ BiCosa4-B2Cos2a+.... 

 + Aj sin a+ Ag sin 2a + .... 



/(a + ^) = B(,+ BiC0s(« + ^) + B2C0s2(« + ^) + .... 

 + Aisin (« + ^) + A2sin 2(a + it) + .... 



