RELATING TO THE BREAKING OF RAILWAY BRIDGES. 715 



tions (33) will give cr (a;) from a; = .28 io a; = .72, and lastly the equation a (x) = o- - x) 

 + (.c - ,v')~-(p (x) will give a (.r) from .r = .74 to a; = 1. 



11. The equation (21) will enable us to express in finite terms the vertical velocity of the 

 body at the centre of the bridge. For according to the notation of Art. 2, the horizontal and 



vertical coordinates of the body are respectively 2c.r and i6Sy, and we have also —^ — = V, 



dt 

 whence, if v be the vertical velocity, we get 



d.lGSijdx %SV , 



But (21) gives/' (^) = \ <p' (g)) whence if v^ be the value of v at the centre, we get 



i-rrSVfi:' SttSVB- 



V, = , or = — ^ ■ , (35) 



ccosrir c {e'"' + e''") ^ ' 



according as /3 < > ^. 



In the extreme cases in which V is infinitely great and infinitely small respectively, it is evi- 

 dent that v^ must vanish, and therefore for some intermediate value of V, v^ must be a maximum. 

 Since V oz (i~i when the same body is made to traverse the same bridge with different velocities, 

 Uj will be a maximum when p or </ is a minimum, where 



p = 2/3-icosr7r, q = (i'^- {^''+ £-<""). 

 Putting for cos rir its expression in a continued product, and replacing r by its expression in 

 terms of /3, we get 



1 - 4/3n / I - i(i\ 



. .f 1 - 4/a\ / I - 4«\ 



whence 



dlogp 11 1 



"d^= "i^ + iT¥^ ^?:7T^"^ ^ ' 



The same expression would have been obtained for ^ . Call the second member of equa- 

 tion (36) F (/3), and let - iV, P be the negative and positive parts respectively of F {(i). When 

 /3 = 0, iV= ec , andP= -+ — -... = l, and therefore F (/3) is negative. When /3 becomes 



infinite, the ratio of P to jV becomes infinite, and therefore F (/3) is positive when /3 is sufficiently 

 large; and F (/3) alters continuously witli fi. Hence the equation F (/3) = must have at least 

 one positive root. But it cannot have more than one ; for the rates of proportionate decrease of 



• • »r ^ I dN I dP 

 the quantities A, /*, or - — — , , are respectively 



^ Ndfi P dfi '^ ■' 



!_ (1 .2 + l3)-'+i2.3 + (iy'+... 

 /3' (1.2 + /i)-' + (2.3 + ^)-' + ...' 



and the several terms of the denominator of the second of these expressions are equal to those of 

 the numerator multiplied by 1 .2 + ft, 2 . 3 + (i,... respectively, and therefore tlie denominator is 

 equal to the numerator multiplied by a quantity greater than 2 + /3, and therefore greater than /3 ; 



so that the value of the expression is less than -. Hence for a given infinitely small increment of 



ft the change - dN in N bears to iV a greater ratio than - dP bears to P, so that when A' is 

 greater than or equal to P it is decreasing more rapidly than P, and tlicreforc after having once 



