1893.] On Operators in Physical Mathematics. 113 



Consideration of a more general Operator, (l-l-v"" 1 )^' Suggested 

 Derived Equivalences. 



33. Some years since, after noticing first the analytical and then 

 later the numerical equivalence of the different formulas for the 

 Fourier-Bessel function arising immediately from the operator 

 (l + V" 1 )"" 4 by the use of the two extreme forms of the binomial 

 theorem (the only forms then known to me), I endeavoured to extend 

 the results by substituting the operator (l + v" 1 )"* which includes 

 the former, and comparing the extreme forms. Thus, calling u the 

 series in ascending powers of v -1 > an( i v the descending series, so 

 that 



u = i + ^-i+^-l) v -* + . . . . , (35) 



= v -, ( ' 1+wv+ !L(!j=i> v *+. . . . ), (36) 



and integrating (with # for operand, as usual when no operand is 

 written), we obtain 



, 



and the suggestion is that these are equivalent. If this equivalence 

 is analytical, and we substitute V" 1 for x and integrate a second time, 

 we obtain 



<> 



and obvious repetitious of the same process lead us to 



which are clearly the cases r = and r = n of the general expres- 

 sion 



x r \n 



VOL. LIV. 



