EQUATION AND ITS TRANSFORMATIONS. 
801 
if and 
if q- 
2i + l 
In the case of 5 =——-, we have (2i-\-l)q —1 = 0, and therefore 
(2t — l)g— 1 = — 2q, (2i—8)q — l = —4:q, ... q~Y — —2iq-, 
also 
iq=±—±q, iq —1 = — \q, so that 
and similarly 
(* - 1 )q ((i ~ 1 )q ~ 1 } = WY- 1 )> &c - 
Thus the ^-coefficient which multiplies the right-hand side of ( 22 ) 
_ (!7 3 -l)(3 3 g 3 -l)(5V-l) ■ ■ ■ |(2i-l)t/-l} _ 
(—) i 8 i q.2q.3q . . . iq 
and, writing the terms on the right-hand side of ( 22 ) in the reverse order, the formula 
becomes 
2 2 
VIZ. 
feb) + " r 1) (§h)+ &a ^ s+ '- 
where q=: 
1 
2t + l 
Treating the formula (23) in the same manner, we find 
— |)-ef X (^) + (3^)% & c.) J » 
where q= — ——- 
1 2i + l 
The right-hand members of these two formulae differ from one another and from 
o 
the last expression in § IY. (art. 19) only by the powers of a which occur as factors in 
the two former expressions. 
MDCCCLXXXI. 5 L 
