540 Proceedings of the Royal Society of Edinburgh. [Sess. 
U x u 2 
. . . U k 
ttj & r 
... 1, 
C m q {k ^ — A.u^ -l - ~Qu.-) 4* . . . . 
+ h% = 
« 2 ^2 
. . . l 2 
• (41) 
tt A-l ^A-l • • 
■ • • h-i 
We may now write 
down in multenions 
many analogies to general 
results above in fictors. 
Thus if p v p 2 . . . . 
p k be h 
given 
independent 
multenions and 
ft = C, 
*PaPs ■ ■ ■ 
• ftC s ^^2 . . 
• • Pk } 
Pz= - 
GmPlPs ■ 
• • • pkGrbhPi 
• • • • Pk | 
■ (42) 
then = Sp 2 |p 2 = . . . . = 1 (_ 
Sp 1 \p 2 =Sp 1 \p s = Sp i \p 3 = . ... = 0) 
L({,Z) = mp,p) = lL(p,p) . . 
In place of (21), (23), (24), (26), (27), (28), (29) we have 
C s (<t> - x)Pi(4> - %)p 2 ■•■•(«£- zjPk-Gl'PiPi ■ ■ ■ ■ Pn = Sh lc >( - x) k 
</>* - h'tf- 1 + + ( - )*#| =(</>- a) A (cf> -i) B =0 
h k ' } = SC s (<£p) <c, y i_c lC7 I p (c, p <4-l!) I 
= (e !) _1 ( [h - c] I ' 
[^] = ^ = (A : !)- 1 0^«0 s (^) w = C«Mte • • • • ^CrViPs • • • • ft ) 
= CgCfiU^U^ .... <f)U k i 
0 = [(/)- z] EE 2A (C) ( - #) fc_c . 
<£-V = ([>- 1] !)- 1 .[^]- 1 .C m p- 1 >.C s r^f)«^ 1 ' 
= [^.]- 1 .SC m ^-«C s r(#)'‘-» 
} 
*-V = (|>- 1] !)-‘.[^']- 1 -C„(^'f)' ii - 1 > C s rp-') ) 
(43) 
(44) 
(45) 
(46) 
(47) 
(48) 
(49) 
(50) 
(51) 
These transformations are effected by first making them for the 
particular case given by 
u, = 1 , 
u„ = l. 
Ta$ • - • • 
for which we have the transforming formula 
C s a (n) = = SST|a' 
(n) 
C m a |n -« = Sj^la'”- 1 ' = WKafci 1 ’ / ' 
ajT>-C.a<*».W, alJ=KC„a'"-«.CT . 
in which it is to be remembered [(41)] that C m «-“ -11 is a fictor. 
(52) 
(53) 
• ( 54 ) 
