INVERSE METHOD OF DEFINITE INTEGRALS. 345 



'-pp- = i /ee'-^o'S {Ao cos (x sin + n9) + A, cos [a; sin d + {n-l)e'\ 

 (toe '^ 



+ ^a cos [ic sin +(» — 2)0] + &c.} 

 + _|ge'^cose |^_jCos[xsine + (re + l)0] + ^_,cos[xsin0 + (w + 2)0] 



+ ^_3 cos [a; sin + (w + 3) 6*] + &c4 . 



I. When n «'* a positive integer, the whole of the second line 

 vanishes, there will then be no arbitrary constants; also, the first n 

 terms of the upper line disappear. 



II. When n is a negative' integer, the first n terms of the second 

 line remain, and these contain n arbitrary constants. 



III. When n is jractional, the whole of the second line remains, 

 giving an infinite number of constants. 



21. The theory of numbers as connected with definite integrals, 

 afibrds another remarkable application of reciprocal functions. 



Let n be any integer of which the divisors are n, Ji', n" 1; also 



let m be any intger, and d an arc of which the limits are 0, tt. 



Then, generally, 



1 -2Acos»?0 + A'' = (l - A6'»e^^)(l- Ae-"*^^); 

 and hence, 

 h. 1. (1 — 2 A cos ra + A") = - 2 { A cos m + ^ ^' cos 2 »J + ^ A^ cos 3 /w + &c. I . 



Suppose now that m is one of the numbers n, n', n" 1; this 



series must contain one term involving cos»0, viz. 



— A^cos w0: 

 n 



and therefore. 



Tit — 



j^cosw^h.l. (1 — 2Acos»»0 + A^) = — TT. — . A"". 



But when m is not a divisor of n, there will be no term in the 

 expansion found to contain the arc n9, and therefore, 

 ^cos«0h.l. (1 -2ACOSJW0 + A^) = 0. 



