. TRANSACTIONS OV SECTION A. 467 



or xff (D)m =/ (D - a )X'U, 



we may generalise thus : — 

 t— i 

 Operate with [D - 1] on both members of (4), then 



[D]Vh. = - U ( J± D - _?L_) [ D f \ 

 o \m — r m — r) 



Write r + n for w, then 



[B]W= (- °r )Vf -=-D - -JL + lT[B-f V, 

 \ o / Lto — r to — r J 



So generally, 



p>]V,h. = (- 4 ) Vr — D — +*-ilW~* 



\ o / L to — r wj — r J 



By applying the same process to (5), or more simply by writing, in (6), p for q 

 and interchanging a, to with b, r, we find 



ptfwi = ( - H )Vr _Z_ D _ JL_ +p -l]\B]- P u n 

 \ a ) L r - to r — m J 



\ n/ Lto — >• to — r J 



Therefore, when qr =pm, 



L»»-r m-r J L J 



=(->s ^- p r — d - -=- +?-i iW" v, (6) 



o</cp Lto-?- TO-r * J L J w 



If we make t = p, and interpret [D] by 1, this equation becomes 



Lto — r m — r-J 



- ( -> 4 is *~r — d - — + 1 -iiW v„ en 



o«c Lto-?- to— r J L J " v ' 



which may be transformed into 



Lm-r m-r-1 J " v ; b« c» Lm-r m-r 2 J "' <■ ' 

 an equation which may be obtained directly from (6) by making t = q. 



When hj and »• are prime to each other, the lowest values of p and q are r and 

 to respectively, and either (7) or (8) may be written in the form 



Lm-r' dx m-r A L dxA W 



_, ™«'c m r to d « .i™ 



- ( - } Syr l^Tx-rr^-r "U f^- < 9 > 



or in the form 



L/ - -to or j--toJ L iiJ 



(_) r ^ r JL_ k« __»_ _! l\,- mu ,, (10) 



«' c m L- to r-»t J v ' 



r — in 



The general integral is 



m» =c 1 y 1 " +c 2 y 2 » •• ■+c,.y r » •• +c m y„,«, 



H H 2 



