24 
ME. W. SPOTTISWOODE AX EXTENDED FOEM OF 
For ViU, 
- (i+i4-8)^V.+ •■} 
1 f 1 A / ^ I ^ \n — 2 + 4 ] 
1 , 1 — 2 + 2 ^ I 
2-2 + • • 
22 — 2 + 2 
2 — 2 ' ( i - 
( n—i + 2 )( n — 2‘ + 4) 1 f 1 
>-2) (2-4) 2-4 
2 — 4 ^'-® (i— 4 ”^ 
1 \2l — 2 + 6 
1 1 
and writing for convenience .. j = j + 
expansion with the numerical coefficients suppressed 
2 / 2—2 
e. the multinomial 
22 - 2 + 2/1 1 y 
- *_2 
^ ( 2 -2) ( 2 -4) \r 7=^ j 
the last term being 
-2’+ 2) (22 — 2 + 4) ..(22 — 1) 
/— (22- 
and 
;i_2)(2-4)..l 
/I _L 1 it 
(, 2 ’ 2 - 2 ” -3’ ) 
Ao when ^ is odd. 
—2 + 2) (22— 2’ + 4).. 22 
(2- 2) (2-4).. 2 
when i is even. 
And from the way in which the coefficients are formed, it is easy to see that in the case 
of V>, 
22— 2' + 2/1 1 ^’”1 
(22 — 2 + 2)(22 — 2 + 4) /I 1 1 A® 
■*“ ( 2 - 2 ) ( 2 - 4 ) \2 ’ i ^4 J 
with conditions for ^ even and * odd, similar to those given in the cases of Vf, V?, 
And since it has been shown above that 
F(Vi)w= 2 jF^( 22 — (^ + 1)237) 
