264 Tlie Rev. B. Bronwin on the Coefficients oj Sines 



Hence if we put u^^ Wgj &c. for the sums of the first and last, 

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

 f{x)i and in like manner Vj, Wgi &c« fo^' the differences of the 

 same, we may replace (3.) and (6.) by 



n \ 2' 



■D 2/ iV Silt {n—i)iit\ 



if 2 be even; I ('''•) 



r> 2/ /tt , S/tt , , (w--l)/7r\ 



^^ = d'^' '°^ 2^ +^^*^^' 2^ +•••• +^|"^' 2^> 



if i be odd. 



This will diminish the labour of numerical computation 

 considerably; and when we know the value of ?i, we may for 

 many values of i effect a further reduction. 



Now let 



\|/(a7) = Aisin A' + A2sin2a? + Ag sin3^+ (B.) 



rJ/(a) = Ai sin a + Agsin 2a + Agsin 3a+.... 

 vI/(a4-A:) = Ai sin (« + A;) + A2sin 2(a + ^") + A^sin S{a + k)-{- .... 

 \I/(a 4- 2^) = Ai sin (a + 2^) + Ag sin 2(a + '2k) + Ag sin 3(a + 2lc) 

 + .... 



• • • • • 



t|/(a + («— 1)^) = Aj sin (a+ (?/— 1)^) + A2sin 2(a+(« — 1)^) 



+ Ag sin 3(a+ (« - 1)^) + .... 



Multiply the first of these by 2 sin /«, the second by 2 sin / 

 (a + /c), the third by 2 sin /(a + 2/?:), &c. and sum as before. 

 The coefficient of A^ will contain the terms 



2 sin pa. sin za= cos (p — i)a— cos (p + i)a, 

 2 sin J3(« + k) sin i{u + A") = cos {p — i){u -\-k)— cos {p + i){u + ^), 

 &c., and therefore 

 22\I/(a + i'/c) sin /(a + i'i) = S Ap cos ( j9 — /) ( « + (n— 1 ) - j 



--^ _SApCos(;)H-?)(« + («-l)-) 



sin (i?-0 - 



. , ..nk 

 sm{p + i)~ 



1 («•) 



