RELATING TO THE BREAKING OF RAILWAY BRIDGES. 711 



for it is easy to form a table of differences as we go along ; and when i becomes considerable, the 

 quantity to be subtracted from ^,.., + A^;_, will consist of only a few figures. 



5. When i becomes indefinitely great, it foUows from (11) or (12) that the relation between 

 the coefficients Jj is given by the equation 



■di - 2^,.., + ^,_2 = 0, (13) 



of which the integral is 



Ai = C + a (14) 



Hence the ratio of consecutive coefficients is ultimately a ratio of equality, and therefore the 

 ratio of the (i + l)th term of the series (9) to the ith is ultimately equal to w. Hence the series is 

 convergent when x lies between the limits - 1 and + l ; and it is only between the limits and 1 

 of X that the integral of (S) is wanted. The degree of convergency of the series will be ultimately 

 the same as in a geometric series whose ratio is x\ 



6. When *• is moderately small, the series (9) converges so rapidly as to give z with little 

 trouble, the coefficients J„ A,... being supposed to have been already calculated, as far as may be 

 necessary, from the formula (12). For larger values, however, it would be necessary to keep in a 

 good many terms, and the labour of calculation might be abridged in the following manner. 



When i is very large, we have seen that equation (12) reduces itself to (13), or to A°^ _.. = 0, 

 or, which is the same, AM,- = 0. When i is large, AM,- will be small; in fact, on substituting 

 in the small term of (12) the value of J; given by (14), we see that A^^,- is of the order i-'. Hence 

 A^^j, A"*^, ... will be of the orders i"^, ^-^.., so that the successive differences of 4,. will rapidly 

 decrease. Suppose i terms of the series (9) to have been calculated directly, and let it be required 

 to find the remainder. We get by finite integration by parts 



2.4;^?' = const. +Ai _ - i\Ai- — + A'J;;^ rs"---. 



w - I {x - \y (a? - 1)^ 



and taking the sum between the limits i and os we get 



J;*' + 4;+.«' + ' + ...to mt = x'-' L.-^+ A^i (-f-.\\ A^f-^] + ... l; ...(1.5) 



z will however presently be made to depend on series so rapidly convergent that it will hardly be 

 worth while to employ the series (15), except in calculating the series (9) for the particular value i 

 of X, which will be found necessary in order to determine a certain arbitrary constant*. 



7. If the constant term in equation (4) be omitted, the equation reduces itself to 



dx' {x — x-y 



The form of this equation suggests that there may be an integral of the form y = x'" {\ - ,r)". 

 Assuming this expression for trial, we get 



(Pv 

 (x - cT-T ~ = x" (\ - x)' {m (ill - 1) (1 - xf - »mnx (1 - x) + n (n - 1) .ti-'J 

 dx' 



= y \m{m - I) - 2m {iii + w - 1) x ^■ (in + n) (m + n - 1) .i''{ . 

 The second member of this equation will be proportional to y, if 



TO + w - I = 0, (17) 



,+ ,..; ", = (16) 



• A mode of calculating the value of z for .c - .5 will prcHcntly be given, which ii easier than that here nientioiied, unlciia ji he very 

 large. Sec equation (12) at the end of thi» pai)tT. 



4 Y 2 



