524 Povceedings of Royal Society of Eclinhurgh. 
This and five other similar substitutions give us 
/3 k Jc\ 12 3 12 3 
P(a ; a, a ) = <^(2 - -2) aaa + <^(3 - l)cf>{2 - 1)<^(2 -3) a a a 
\3 /« /</ 12 3 13 2 
12 3 12 3 
+ (^(1 - 2)<jS(3 - 2)c/>(3 -1) aaa + cf>{3- 2)(^(1 - 2)(^(1 -3) aaa 
2 1 3 2 3 1 
12 3 12 3 
+ <^(1 - 3)cj5>(2 - 3)</>(2 -l)aaa +c}>{2- 3)cj>{l - 3)cjf)(l -2) aaa- 
3 1 2 3 2 1 
SO that the law is seen to hold also for the case of three permutable 
indices. The completion of the proof, giving the transition from 
n to + 1 jiermutable indices, occupies three pages. 
This is followed by two pages devoted to the subjects temporarily 
set aside at the outset, viz., the possible existence of functions 
having the peculiar properties of <^. Two amusing instances of 
such functions are given, — 
(1) 
( 2 ) 
■ ■ ■ 
p^-l _ p^-2 _ p^_3 _ 
■^0 -^0 0 
sin2fa sin 2/377 sin 
‘PyP >- {ji - ■ ■ ■ 
sin 2/3^ sin 2/3tt sin 2^7 t 
“(/3+l)2;r“(/3+2)2,r~(;8 + 3)27r“- ' ’ 
2 
+ . . 
,sin 2/37T 
where Vf stands for the coefficient in the expansion of {a + b)^ . 
Success, however far from brilliant it may be, in thus expressing 
the rule of signs by means of the symbols of analyses, led Sclierk 
to try to do the same for the rule of formation of the terms. 
xTothing came of the attempt, however. “ Bald aber,” he says, 
zeigte es sich dass Permutationen niemals durch andere analytische 
Zeichen ersetzt werden konnten.” 
Such speculations are not altogether uninteresting when later 
work like Hankel’s comes to be considered. 
In an Appendix dealing (1) with the case of a set of linear 
equations which are not all independent, (2) with the solution of 
particular sets of equations, there is given at the outset a proof of 
