Ross — Verb-Functions. 



57 



19. Remarks. — (1) The coefficients produced by may be stated 

 in yarious ways, and can easily be tabulated ; itself may be replaced 

 by a series of differential operators ; and the two whole series may be 

 recast in several fonns. It is impossible to examine these details here ; 

 but it should be noted that 



= - (?) [/5" + V-iP"'y + + 2^-2/8"-']-' + [iS" + p-3i8""']-' + etc. 



+ terms involving compounds of ^.o, . . . ; 

 and also that it 



= y8» |l - ^ log ^1 + + . . . j + terms divided by n\2, . . .| . 



(2) The general equation \_f^x = y is dealt with by expanding 

 \_f~\x in powers of x or [<^]^, and then inverting the expansion by 

 operative division. Thus, 



[/„+/'.. ^ + ...]-' = /3-^/3^ + ..., 



B -f 



where the ^ of the invert operates on - • 



/ 



(3) It must be remembered that, though the subject of an operation 

 be unreal, the result need not be so ; for example, 



[/3^ - 3/3]-' (- 2) = [x] y~2 = ± + f . 



20. Superior Division and Synthetic Division. — These processes 

 may be briefly referred to here, as they help to demonstrate the fact 

 that the results arrived at above are of the nature of perfect identities. 

 If ^ = [^]X' then the value of that is, [i/']"^<^ can be found by 

 superior division without first finding the value The process is 

 similar to that of inferior division given above, except that the whole 

 divisor, i/^, now operates on the whole quotient. As each subtrahend 

 is formed, terms already used in previous subtrahends are omitted. 

 Of course i}/ \ fS = jS \f/. Synthetic division may be employed to 

 obtain an invert without having recourse to expansions of multi- 

 nomials raised to fractional powers. For suppose we requii-e the 

 invert of 



^" + afS""-' + h(3"-' . . . 



