Product of a Special 'N-Line Determinant. 
275 
and that in the first two columns all of them are identical. Not much 
more is then wanted to enable us to suggest the construction of the 8-line 
determinant 
/3x ft. 
yi 72 
73 
75 
74 73 72 
ft, ftz ft^ 
C4 
• 74 ^75 
. ft, ^fts 
Xa^ ttj |, 
of which Laplace's expansion-theorem gives the equivalent 
I "1/3273^4 1' - 1^273^5!"^ + \a^ft^y^^A\'\(i.ft.y,^A>< 
as desired. 
It is a help here to note {a) in the first four rows of the 8-line deter- 
minant the repetition of the column with the suffix 4 ; {h) the mode of 
obtaining the last four rows from the first four ; (c) the fact that but for 
the X's the determinant would be centrosymmetric. 
3. That the result thus obtained includes Mr. Eoseveare's is readily- 
shown. As an example, let us take the next lower case of the extended 
theorem, namely, 
«x/3273l' - l«A74l'^ + l«x/3273l-|«A74l^ 
Making in this the substitution 
ft, 
73 
ft, fts 
74 73 
73 72 
^ft, ft, ft. 
/5x ft. ft, 
7i 72 73 
-^5X3 
-^^,X3 
- a^X^ 
tti - a^X^ 
ao - aX" 
a^ - a^X 
a^ - a^X 
- a A 
we of course at once find the left-hand side identical with one side of Mr» 
Eoseveare's, and the right-hand side changed into 
tto a^ — a^X^ ttj — (X5X a^ — a^X"" 
a^ — a^X^ a^ — a^X^ a^ — a^X a^ — a^^ 
— %X4 —^5X3 a^ — a^X'^ a^ — a^ a^ — a^X^ 
a^X — a^X^ ao — a^X^ —a^X^ —a^X"^ 
a^X — a^X^ a^ — a^X'' ao — a^X^ 
a^X — a^X^ a^ — a^X^ a^ a^ 
21 
