7 60 Proceedings of Royal Society of Edinburgh. [sess. 
a , 
“ Let the symbols 
a, ft 
a', P' 
a > P, 7 
a , P\ y 
tt 
a 
,P',y" 
&C. 
(VII. 10) 
denote the quantities 
a P' - a'P , ap'y" - a/3"y' + a P"y - a Py" + a ' py - a "P'y , &C. 
the law of whose formation is tolerably well known, but may 
be thus expressed, 
a , p 
a , P 
= a\P'\ 
- a \P\, 
a } P , y 
a, P’, y 
tt r\tt tt 
a y P , y 
p. 
y 
\P'> y" 
+ a 
+ a" 
A 7 
/3", 
tt 
7 
1 ft 7 
P> y 
&C. 
the signs + being used when the number of terms in the side 
of the square is odd, and + and - alternately when it is even. 
Then the theorem in question is 
p a + a P + t y ... y p a' + c j8' + T y ... , p a" + v P" + r y" . . . 
p a + cr P + t y . . . , p a + or' P' + r y . ■ . , p a" + <j P" + r y" ... 
p a + (r p + ry...j pa+crp+ry pa +c r p +r y ... 
p y <T'y T ... 
a , p y 7 ••• 
Py O-'y T ... 
a , P'y y ... 
P'y V", T" ... 
a", P", y" ••• 
“ This theorem admits of a generalisation which we shall not 
have occasion to make use of, and which therefore we may 
notice at another opportunity.” 
