452 Proceedings of the Royal Society of Edinburgh. [Sess. 
8. As a first step towards arriving at the final expansion of the 
function, we take note from the original determinant form that the 
aggregate of the terms containing h 
a + c + d + 6 
a + d 
a + c 
a + d 
a+b+d+f 
a + b 
a + c 
a + h 
a+b+c+g 
a+c+d+e 
c + e 
d + e 
c + e 
b+c+e+f 
b + e 
d + e 
b + e 
b + d + e + g 
and consequently that the aggregate of the terms containing ha is 
ha | (b + e)(c+f) + (c +f)(d + g) + (d + g)(J> + e)^ - 
We then, from the same source, obtain the terms containing li without a, 
namely, 
4 bed + ( cd + db + bc)(e + /+ g) 
+ (c + d)fg + (d + b)eg + (b + c)ef+ efg ; 
thence, by means of the interchange (VII.), the terms containing a with- 
out h\ and finally, by recurring to the original determinant, the terms 
independent of both h and a , namely, 
be(c +f)(d + g) + cf{d + g)(b + e) + dg(b + e)(c+f ) . 
The result thus reached, 
0 o 
2 ,( ha + be)(c+f)(b + g ) + ^(a + h)be(c+f+ d + g) 
+ a(bcd + iefg + ^bfg) + h(efg + 4bcd + ^ ecd), 
(VIII.) 
though not elegant, is useful for purposes of verification. 
9. Having, however, obtained the terms in ha, we are led to use the 
first three forms of B in § 6 to arrive at three other aggregates of like 
constitution ; and a little additional trouble results in the complete 
expression 
ah\ (& + e)(c+/) + (c+f)(d + g) + (d + g)(b + e) | + (a + h)(efg + bed) 
+ be | (c+f)(d + g) + ( d + g)(a + h ) + ( a + h)(c+f ) i + (b + e)(agf + hdc) 
ax.) 
+ cf <j (d + g)(a + h) + (a + h)(b + e) + (6 + e)(c? + ^)| + (c+f)(gae + dhb) 
+ dg ^ (a + ti)(b + e) + (b + e)(c -\-f) + (c +f)(a + h) j- + ( d + g)(fea + cbh ) 
which, until we reach the last binomial in each line, is a function of 
ah , be , cf , dg and a + h , b + e , c+f , d + g . 
