54 Proceedings of the Royal Irish Academy. 



etc. 



1 i2 



w 















L2 



(r.) 



L2r 



/5' 



-3U^-^^y — [2 — -jT- 



_ 1 J^Y'^ ^P-iP-i+^P-2P-^ , 3p-^'p.^+Sp.,p.,' 



4 y [2 y |_3 



|_4 \^.^y 



= - P.l-P.2l^'^- {P-Z+P-lP-2) P~'-{P-i + 2p_,p.^+pJ+P_l'^p,o)/3-^ 



- { ^-5 + ^P-1 P-i + 3i?-2 p.3 + ^P-i^P-z + ^P-iP-2^ + P-iP-i}!^'^- etc. 

 It will, of course, be understood that 



(m)^**^ = 7)1 {m - \) . . . (7)1 - r + 1), 



and that 



(_ = (- l/m + 1) . . . + r - 1). 



Hence, operative inverts can be quickly obtained by the following 

 rules : — 



(1) Write out the algebraic invert of 



either by algebraic division, or by the multinomial theorem; if the 

 latter be selected, expand 



and rearrange, if necessary, in terms of /?. 



