344 Lord Rayleigh : Problem of Random Vibrations, 

 we find for the various ranges : 



(r<2l), -20; (2/<r<4/), +10; 

 (4/<r<6/), -2; (670), 0. 



And on integration for 



-**(?%)> < 64) 



(0-2/) -^+4/r 3 -lG/V, 



or 



(2/-4Z) + v> -6/r 3 + 30ZV-56ZV+20/ 4 , 



(47-6/) - ^ + 2/r 3 - 18/ V + 72/V-108/ 4 , 

 (r>(;/) 0. 



AVe may now seek the form approximated to when n is. 

 very great. Setting for brevity / = 1 in (59), we have 



, / sin .A" f x 2 -1 



Io 4 v ) -»■[— g+^+^+.-.j, 



where 



""180' " 6 ~~35~81 



. 1 . 1 



^-~Tx7v A «=-air«T' • • • ( 65 ) 



and 



( ^ )" = ''"" ,; "< i + "**** + "**• + K*«v+ • ■ • } , 



so that 



-^=-1 xdxsmrxe- nx2l *{l + n\x*+nhy 

 r ar ttJq 



+ in > A 4 V +■•»}. (66) 



The expression for the principal term is a known definite 

 integral, and we obtain for it 



dFn _ 3y/6 . V 2 -3r2/2n ™* 



dr ^/nr.n^ e ' ' * ' ' ^ } 



which may be regarded as the approximate value when n is 

 very large. To restore /, we have merely to write r\l for r 

 throughout. 



In pursuing the approximation we have to consider the 

 relative order of the various terms. Taking nx 2 as standard, 

 so that x 2 is regarded as of the order l/?i, nx s is of order n~ z 

 and is omitted. But n 2 x % is of order n~ 2 and is retained. The 



