{t''"') 



310 Sir William Rowan Hamilton on Fluctuating Functions. 



/, = i- ^ 1™ f de sin e'\ da sin (2^ (a - ^) sin 6} (a - x)-'f ; (s''^^^ 



a theorem which may be easily proved a posteriori, by the principles of fluctua- 

 ting functions, because those principles show, that (if x be comprised between the 

 limits of integration) the limit relative to /3 of the integral relative to a, in (s^^"), 

 is equal to Ttf^. In like manner, the theorem (c), when applied to the present 

 form of the function p, gives the following other expression for the arbitrary 

 function/", : 



^ rj> ^ do sin 6 sin (2 (a — x) sin 6^ cos (An (a — x) sin 6^ ; 



+ (n)^)^ «/a 5; de sin e sin (2 (a — x) sin o) 



X being between a and b, and b — a being not greater than the least positive root 

 V of the equation 



- C rfe sin sin (2 V sin e) = 0. - (u """ ) 



And if we wish to prove, a posteriori, this theorem of transformation (t'^"), by 

 the same principles of fluctuating functions, we have only to observe that 



1+22," cos 2ny = !!^^±i^), (v-) 



and therefore that the second member of (t*^^^^) may be put under the form 



iirv, p' ^"f/Csin 6sin ('(4re + 2) (a — ^)sin0^ 



1™ i daf— ^— ■ — — — _— _Z . (vf'^'") 



n=ccJa •^" 2 5^ rfe sine sin (2 (a — a:) sine) ' ^ ^ 



in which the presence of the fluctuating factor 



am (^{An -\- 2) (a — a;) sine), 



combined with the condition that a — a; is numerically less than the least root of 

 the equation (u^^"), shows that we need only attend to values of a indefinitely 

 near to x, and may therefore write in the denominator, 



C de sin e sin (2 (a — x) sin e') = tt (a — x) ; (x''"') 



