REPORT ON CERTAIN BRANCHES OF ANALYSIS. 251 



p(iogO) _ Q^ yfQ jyjay replace the preceding product by the equi- 

 valent expression 



This expression, which is equivalent to ^Db" '■ j, 



has been applied by Libri to the expression of many important 

 theorems in the theory of numbers *. 



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



It 

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



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

 follows, therefore, that 



2 r^dr . (b-a) C (a + b)l 



— / — sm ^^ — Tz — - r cos ■< x — ^^ — 7^ — - r > 



vJo r 2 L 2 J 



= — / ^ — sin (^ — a) r H / — sin (^ — 6) r 



vJo r ^ ' TTi/o »* 



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 



definite integral — / — sin — - — - r cos -s x — ^ — -- — '- > r 

 ^ -K JQ r a L 2J 



by Cj", we shall get, 



xT\ « n«i ^ ^ 1 " — b 



^' -^* +2(a:-a) + 2]F=T)' 



and consequently the equation of a polylateral curve, such as 

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



y = Cb . <P]X -{■ C/ . 9.2 -^ + Cd . ^3 X +&C., 



in as much as at the limits we have <Pi {b) = p^ (b), (p^ (c) s= a^ (<?), 



and consequently for such limits Cj" ^j (b) + C/ f^ (^) = f 1 (^) 

 = ip^ (b), 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 



• Cr elk's Journal for 1830, p. 67. 



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

 loiii, Memorie della Societa Italiana, torn. xx. p. 448. ; Libri, Memoires de Ma- 

 thematique et de Physique, p. 40. 



