274 THE KKV. F. H. JACKSON 



Similarly, from (55)-(56a) we obtain 



y [y) = v o( x ) J m{ vx ) + 2v i( x ) J d vx ) + 2 M x ) J [d vx ) + » • • ( 71 ) 



"»( ?) = 2 | "'■(?) J [«+r]( v ») + "»+r(a')JM(^) [ , 



K( x )=Ky}) J U i ' x ) + 2k i(jj J m( ux )+ 



K( X ) = 2, { X 'Q) J Cn+-'j( M: '') + ^wQJmC"*) > , 



in which 



ag 2 hq 1 



1 _ ? 2 ' ! _ ? 2 



Part III. 



Certain a -Equations and their Limiting Forms as Differential Equations. 



§ 1. The General Hypergeometric Series. 



It will only be possible in this section of the paper to state the forms of a few 

 types of A -equations and their solutions, without entering into very detailed analysis. 



<l>[xA.y]- ±i/,[xAy] = (72) 



has an infinite number of solutions of the form 



y = A { *>+ *M r r"+i + *Wf« +1L «.+2 + I, . . (73) 



in which 



<£[a:.A t y] denotes Ctf/ + G 1 z.&y + C 2 x 2 A' 2 y + .... + C> 5 A 5 ?/, 

 i/{a;.Ay] „ C' y + 0\xAy + C> 2 A 2 s/ + .... +C t x t A'y, 



for, on operating with (p\_x. A 2/] on a series Vc r aj 0+ '', 

 we obtain 



2« a+r «r{C + C 1 [a + r] + C 2 [a + r][a + r-l]+ . . . + C,[a + r] . . [a + r - s + 1] } = ^c^+^a + r] . 



Similarly operating with - \|/-[sc. A?/] we obtain ^c r x a+r ~ 1 \|/[a + ?'~|, so that the 



solutions are furnished by the function (73). in which a is any root of the indicial 



equation 



C' + C\[a+r] + V'„[a-{-r][a+r- 1]+ . . . +C',[a + r] . . . [a+r-t+l]. 



The general Hypergeometric Series of ^-form is a special case of this equation and 

 depends on the following theorem, which may be proved by induction : — 



« ««•*-] -«• (^X^ 1 ) • • • • (^> 



then II„[a + m] = A + A 1 [m] + A 2 [m][m- 1] + .... + A„[m][m-1] . . . [m-n+1], 



