Sir William Rowan Hamilton on Fluctuating Functions. 277 



sin j: 



X 



0. 



[8.] The known theorem just alluded to, namely, that the definite integral 

 (r') becomes = tt, when n,, := sin a, may be demonstrated in the following man- 

 ner. Let 



c" , sin So 

 A = V da i— ; 



C" , cos /3a 

 B = Wa T-r^ ; 

 J« 1 + a^ 



+ ' 

 then these two definite integrals are connected with each other by the relation 



^=(S/^-i)«' 



because 



C^ 1^ C 1 sin /3a 

 V rf/3B = \ da l" , 



d c" 1 a sin /3a 



and all these integrals, by the principles of the foregoing articles, receive deter- 

 mined and finite (that is, not infinite) values, whatever finite or infinite value 

 may be assigned to /3. But for all values of /3 > 0, the value of a is constant ; 

 therefore, for all such values of /3, the relation between a and b gives, by inte- 

 gration, 



e-^ 1(5 <;/3 + l) B — a1 = const. ; 



and this constant must be = 0, because the factor of e~^ does not tend to become 

 infinite with ^. That factor is therefore itself = 0, so that we have 



A = (^''rf^+l)B, if^>0. 



Comparing the two expressions for a, we find 



B + ^B = 0, if^>0; 



