1893.] On Operators in Physical Mathematics. 515 



the well-known formula for I (a) in rising powers of the square of 

 the variable, as required. 



Transformation to a Descending Series. 



14. There is such a perfect harmony in all the above transforma- 

 tions, without a single hitch, that you are tempted at first to think that 

 you may do whatever you like with the operators in the way of alge- 

 braical transformation. There is a considerable amount of truth in 

 this, but it is not wholly true. I shall show later some far more 

 comprehensive and surprising transformations effected by simple 

 means. At the same time I should emphasize the necessity of 

 caution and of frequent verification, for no matter how sweetly the 

 algebraical treatment of operators may work sometimes, it is subject 

 at other times (owing to our ignorance) to the most flagrant failures. 



But in the above we only utilized one way of effecting the binomial 

 expansion., There is a second way, viz., in ascending powers of the 

 differentiator. The two forms are algebraically equivalent so far as 

 the convergency allows, but we have, so far, no reason to suppose 

 that they are analytically equivalent. But on examination we find 

 that they are. Thus, using the first of (25) and expanding, we 

 get 



(30) 



Here the operand is t or unity. Or we may make it (jp/2a)* if we 

 please. If we know its value, as a function of t, the rest of the work 

 is easy, as it consists merely of differentiations. But nothing that 

 has gone before gives any information as to the meaning of p*, let 

 alone its value. We may, however, find it indirectly. We may 

 prove independently that when at is very big, I (a) tends to be re- 

 presented by e a *(27rat)-*. From this we conclude that the value of 

 p 4 must be (vt)~*. Then (30) becomes 



!(*) = e-> (l--+ (-}'-.. ) L.J , (30a) 



\ 4 a 2 \4aJ )(2irat) k 



and now performing the rest of the differentiations, we arrive at 



which, on test, is found to be equivalent to the ascending series (29). 

 Of course only the convergent part of the series can be utilized for 



VOL. Lll. 2 M 



