RELATING TO THE BREAKING OF RAILWAY BRIDGES. 713 



At the limits x = and x = 1, we have w = cc and w = l, « = co and s = 0, whence if /denote 

 the definite integral, 



We get by integration by parts 



r s'"ds s-" m rs^-'ds 



J + sf " 2(1 +sy "*" i" J (1 + sf' 

 and again by a formula of reduction 



Now fi being essentially positive, the roots of the quadratic (18) are either real, and comprised 

 between and I, or else imaginary with a real part equal to ^. In either case the expressions 

 which are free from the integral sign vanish at the limits s = and s = oo , and we have therefore, 

 on replacing m (1 — m) hy its value /3, 



^^^ roo s"'-'ds 

 2 J^ 1 + s ■ 



The function (p (x) will have different forms according as the roots of (18) are real or imaginary. 

 First suppose the roots real, and let ?» = A + r, w = 1 — r, so that 



r = yTZr^. (23) 



In this case /» is a real quantity lying between and 1, and we have therefore by a known formula 



fCCs'"~'ds TV -K , ^ 



I = -. = , (2*) 



Jo 1 +« sinniTT cosrTT 



whence we get from (22), observing that the two definite integrals in this equation are equal to each 

 other, 



^(.,= ^z_^^^J(^y. (^y] (25) 



' 4rc()srir t.\l - x] \1 - xl ) 



This result might have been obtained somewhat more readily by means of the properties of the 

 first and second Eulerian integrals. 



When /3 becomes equal to 1, r vanishes, the expression for (p{x) takes the form JJ, and we 

 easily find 



^W = J\/.i^-«''l°g7T^ ^^"^^ 



When (i>\, the roots of (18) become imaginary, and r becomes p \/ - \, where 

 ^=v//3^ (27) 



The formula (25) becomes 



(f)(x) = \/x - x" sin n log 



^^ ^ pie"" + e-"')^ V ® 1 -*; 



(28) 



If /(,r) be calculated from a? = to a? = ^, equation (21) will enable us to calculate it readily 

 from X •= ^ to «• = 1, since it is easy to calculate (p^v). 



9. A series of a simple form, which is more ra])idly convergent than (i)) when .i a|)i)r()aciics 

 the value J, may readily be investigated. 



