314 Sir William Rowan Hamilton on Fluctuating Functions. 



[0]-' = 1-1. 2-'. 3-'. 



1 . 



[-i-r = -7 



1—3—5 1— 2^■ 



(n«) 



2 ' 2 2 ' 2 ■ . 

 For the value ^ = 20, the 3 first terms of the series (m^^) give 



9 \ cos 86°49'52" , 1 sin 86°49'52' 



•^^—\} 204800 J 



(o«) 



204800; ■/20^ ' 320 x/^Q^ 



= 0,0069736 + 0,0003936 = + 0,0073672. 



For the same value of j3, the sum of the first sixty terms of the ultimately con- 

 vergent series 



/.=Vo([or)*(-/3')' (p") 



gives 



/,o = + 7 447 387 396 709 949,9657957 t 



- 7 447 387 396 709 949,9584289 



J 



(q^^) 



= + 0,0073668 



The two expressions (m^^) (p^^) therefore agree, and we may conclude that the 

 following numerical value is very nearly correct : 



-'{do, cos (40 sin a) = -\- 0,007367- (r") 



[27.] Resuming the rigorous equation (w), and observing that 



we easily see that in calculating the definite integral 



in which the function f is finite, it is sufiicient to attend to those values of a. 

 which are not only between the limits a and h, but are also very nearly equal to 

 real roots or of the equation 



7x = 0. (U-) 



The part of the integral (t"), corresponding to values of a in the neighbour- 

 hood of any one such root x, between the above-mentioned limits, is equal to the 

 product 



