778 Dr. J. W. Nicholson on 



The next formula obtained in the same manner is 



p r> n p ^ ^ m — 1 • ^ m ~~ ^ • ^ m ~~ ^ . 2??i — 7 

 . 7 ??i . ??i — 1 . m — 2 . m — o 



the work becoming laborious at this point. But the nature 

 of the final result is already clear, and a proof by induction 

 can at once be given. If we suppose that, when m > ft, 



v> r\ c\ T3 cs \ 2m — 1 . 2m — 3 . ... 2m — 2ft — 1 



Km Hn — Hm ^n =/ (ft) . ; „ ~ */ », 



m.m — l.m— Z.... m — ?i 



where y n is the series denoted by Series C, and /(ft) is a 

 function only of ft, and not m, we can show, by use of the 

 fact that 



= (2ft + lHP m (^-Q TO P n )-ft(P^Q„. 1 -^P w _i) 

 that the formula is true universally, and that 



This latter relation is also clearly satisfied by the initial 

 values, for 



/(0) = 1, /(1)=1, /(2)=g^; /(3)=5 L 7 . 



which are perhaps in themselves sufficient to suggest the 

 relation. Finally, if m > n + 1, 



P n r\ j> —ff ^ 2m — l . 2??i — 3 .... 2m— 2/2 — 1 

 r m i4n-i4 m ±n-JW m>m _ 1>m _ 2 . ...m— n yn ' 

 where 



_ft.n-l.ft-2 .ft-3. ... 1/(0) _ ft! 



2n+l. 2ft-1.2ft-3. ? ..3 3.5. ...2ft + 1 



2 w (ft !) 2 

 2ft + l!' 



-t^W?* ^J??i-tn 



2 n (ft !) 2 m — ft - 1 ! 2ml m — n-1 ! # 



ry * * " 2?z+l ! m! 2?n-2n-2l 2 ?l+1 m ! ; 



