256 Proceedings of Boy al Society of Edinburgh. [sess. 
It follows therefore by addition that the aggregate of terms inde- 
pendent of a 4 in N(a 15 a 2 ,a 3 ,a 4 ,a 5 ) is 
- 2 a 2 + 2«2 a 3 ~ 2«2 a 3 a 4 + 2 ala 3 a 4 a 5 , ( v 2 ) 
and that 
N(a r a 2 ,a 3 ,a 4 ,a 5 ) = a 1 D(a 2 ,a 3 ,a 4 ,a 5 ) - + 2 a^a 3 (v 3 ) 
- 2a|a 3 a 4 + 2a|a 3 a 4 a 5 . 
(12) The general theorem of which this is a case may he estab- 
lished by so-called ‘mathematical induction.’ Subtracting the 
first row of /( a 2 ,a 3 ,a 4 ,a 5 ) from the last row, we have 
/(a 2 , a 3 ,a 4 ,a 5 ) = 
• 
a 2 
a 3 
a 4 
a 5 
a 2 
1 
a 3 
a 4 
a 5 
a 3 
a 2 
1 
a 4 
a 5 
a 4 
a 2 
a 3 
1 
a 5 
a 5 
• 
• 
. 1 
- a 5 
J 
L - 
a 5 )/( a 2> a 3> 
a 4 ) 
+ «5 
a 2 
a 2 
a 2 
a 3 a 4 1 
«3 a 4 1 
1 a 4 1 
a 3 1 1 j 
= (l~a 5 )/(a 2 , a 3 , a 4 ) - al(l-a 2 )(l — a 3 )(l - a 4 ) . 
If therefore the law in regard to /( ) hold in the case of the 
third order, that is to say, if 
/(a 2 ,a 3 .a 4 ) = - 2 3 a£ + 2 3 ai|a 3 - 2 3 a^a 3 a 4 , 
— and this is easily verified — we shall have 
f{ a 2> a 3> a 45 a 5) == — 2 3 tt 2 + 2j 3 a 2 <* 3 — 2 3 a 2 a 3 a 4 
+ a 5 2 3 a 1 - a 5 2 g ala 3 + a 5 2 3 a 2 a 3 a 4 
— a 2 + ag2 3 a 2 — a52 3 a 2 a 3 + a?2 3 a 2 a 3 a 4 
= 2)4^ -f- 2^4 ci 2 a 3 2^4ft 2 c t 3 0'4 "b 2 4 a 2 a 3 a 4 a 5 , 
which shows that it will hold also for the fourth order. 
(13) The expansion of N in terms of simple symmetric functions 
having thus been obtained, the number of different kinds of terms 
