300 Proceedings of Royal Society of Edinburgh, [april 16, 
and further by using for M + d'^ the column is put in 
the form 
a{ta%^-a^%a^ + ah) 
b{'la%^-h^^a^^b^) 
e{ta%^ - + c^) 
d{%a%^-d^ta^ + d^). 
This leads to the transformation of the determinant into the aggre- 
gate of three determinants, of which the first 
= 1 1 = 0, 
the second 
= - :§a4 I a%'^cH^ I , 
and the third 
= I a%~^cW ! . 
If the result be desired in terms of simple alternants alone, the mul- 
tiplication of I a^h^e^^d^ | by is performed by the rule given in 
my Theory of Determinants^ § 129. The product obtained would be 
- -f \a%'^cH^\ - \a}h‘^cH^\ , 
so that the given determinant would be found 
= \a%'^cH^^\ - \a}h^cH^\ . 
On looking back at the process, it will be readily seen that the 
essential step consists in the transformation of f{a),'Eib,c,d) into 
an expression involving a and symmetric functions of all the 
variables a, &, c, d. 
(3) There is another mode of treatment which might occur to 
one who knew the multiplication-theorem above referred to, and 
which is more worthy of illustration, because it immediately leads 
up to the best method of all. Taking the same determinant as 
before, we expand it in terms of the elements of the 4th column 
and their complementary minors, the result manifestly being 
- + h^d^ -}- c^d^)\Wc^d‘^\ -1- b{c'^d^ -1- c^a'^ + d^a^)\a^cM‘^\ 
- e{Pa^ -f dH)^ -i- a‘^h'^)\a%'^d^\ + d{a%^ + + b^c'^y\a%'^c‘^\ . 
Here there are to be performed three repetitions of the same 
multiplication, viz., the multiplication of the alternant | h^c^d‘^ | by a 
symmetric function of its variables, b'^d^ + bH^ cH‘^. Doing this 
we obtain the aggregate of the four products in the form 
