146 EVALUATION OF CERTAIN DEFINITE INTEGRALS. 



Equation (3) admits of several remarkable applications. 

 Thus let us suppose n = 3 and fx = sin ax sin bx sin ex : then 



fx = J {sin (a + b + c) a; sin ( a + b + c) x sin (a - & + c) a? 



sin (a + 5 c)a:}. 

 Consequently, we have by (3), 



sin ao? sin foe sin <%c , 7r_, _ N- 



where, as in trigonometrical formulas, 

 25 = a + b + c : 



the upper sign is to be taken when the quantity to which it is 

 affixed is > 0. 



T ., . /""sin ax sin bx sin ex , TT ,.,_.,_.,_., N 

 Likewise I - ax = - (1 + 1 + 1 + 1), 



J & 



where the signs follow the same rule as in the former case; 

 the different unities involved being the zero powers of s, 

 s a, &c. 



Let us now suppose that fx = sm m x cos zx ; the correspond- 



. , , f 00 sm m x coszx j -At. it r 



ing integral, viz. j ^ - ao; occurs in the theory oi 



J Q x 



probabilities. Its value is given at p. 170 of Laplace's Theorie 

 des Prolabitites, where it is obtained by a method founded on 

 a transition from real to imaginary quantities. The nature of 

 what are called imaginary quantities is certainly better under- 

 stood than it was some time since ; but it seems to have been 

 the opinion of Poisson, as well as of Laplace himself, that 

 results thus obtained require confirmation. In this view I 

 confess I do not acquiesce; but if only in deference to their 

 authority, it may be desirable to show how readily imaginary 

 quantities may be avoided in estimating the value of the integral 

 in question. 



sin m # = ^ -jcos mx - cos (m 2) x + &c.i if m is odd, 



and = + j jsin mx sin (m 2) x + &c. ! if it is even. 



TT T. /o\ f sin m a; cos zx , 



Hence, by (3), | - - - dx = 



J o ^ 



)"-' (-*-2)"' J+&CJ 





