REPORT ON CERTAIN BRANCHES OF ANALYSIS. 251 



gOogo) _ 0^ y^Q j^ay replace the preceding product by the equi- 

 valent expression 



This expression, which is equivalent to ■^Di" j, 



has been applied by Libri to the expression of many important 



theorems in the theory of numbers *. 



f*'^ d r 

 The definite integral / — sin r x has been shown by 



Eulerf and many other writers, to be equal to -^ when x is 



— It 



positive, to when x is 0, and to —^ when x is negative. It 

 follows, therefore, that 



2 r^dr . {b-a) r (« + 6) 1 



— / — sm ^ — rz — - r cos < x — - — -^ — - r > 

 vJq r 2 L 2 J 



= — / ^ — sm (.r — a) r H / — ?,m.{x — h)r 



kJq r ^ ^ "JfJo r ^ ' 



is equal to 1 , when x is between the limits a and 6, to -^, when x is 



at those limits, and to zero, for all other values. If we denote the 



A ^ ; ■ , .^r'^dr.ib^a) r {a+b)\ 



definite integral — / — sm — - — - r cos ■{ x — ^ — ;- — ^ > r 

 ^ -ajQ r 2 L 2J 



by Cj", we shall get, 



aT\<'_ra.a — n . b ~ b 



iJb — ^b + T^ry- r + 



2{x -a)^ 2ix -by 



and consequently the equation of a polylateral curve, such as 

 that which is expressed by equation (4.), will be, 



y = Cb . <P]X + C/ . 02 X + C/ . <p^x + &c., 

 in as much as at the limits we have <p-^ {b) = ^ (b), <p^ (c) :*= ^3 (c), 

 and consequently for such limits Cj" f^ (b) + C/ f^ (b) = ^j (b) 

 = ?2 (*)j and not 2 p^ (b). 



All definite integrals which have determinate values within 

 given limits of a variable not involved in the integral sign, may 

 be converted into formulae which will be equal to 1 within those 



• Crelle's Journal for 1830, p. 67. 



t Inst. Calc. Integ., torn. iv. ; Fourier, Tkeorii de la Chaleur, p. 442. ; Frul- 

 loni, Memorie della Societd Italiana, torn. xx. p. 448. ; Libri, Memoires de Mar 

 thematique et de Physique, p. 40. 



