TJUANS^CTIONS OF SECTIOX A. 597 



then 



J _„(:) = - ( - j {cos 7nr . F(p) + sin mr . /(/>V, 

 •when >^ and s are nearly equal. These lead to the series, when p is small, 



^■<=>-i(!V{ "-a) -i-(«'-i)n^>(i)-i- i"'-'f)*-- • ^ 



The Bessel functions, whose argument is purely imapiiiai v, are usually 

 defined by 



the latter being zero at infinity. 



These have the expansions -when either w or s is larofe. 





,.~t 



where if ji=.- sinh /3 



■where D = 



r = sech (8 - Xj sech" + X^ sech^ 13 . , . 



t = r.(l-!^l)'fijl)-'. . .){cosh^-^sinh^) 



1 d 



sinh cosh /3 ' W^ 



3. 0>i <S'«/' W. R. Hamillon's Fluctuating Functions. 

 By E. W. HoBSON, Sc.D., F.B.S. 



This paper dealt with the mode of representation of an arbitrary function 

 by means of definite integrals involving the fluctuating functions just intro- 

 duced by Hamilton in his memoir ' On Fluctuating Functions,' published in the 

 ' Transactions of the Eoyal Irish Academy ' in the year 1843. The results obtained 

 by Hamilton were discussed in a rigorous manner,' and were extended to the case 

 in which the function to be represented is of the very general type limited only by 

 the postulation that the function possesses a Lebesque integral in the interval fo"r 

 which it is to be represented. General theorems are obtained containing sufficient 

 conditions for the convergence of the integrals at points in the limited or 

 unlimited interval of representation, and also sufficient conditions for the uniform 



