UNDER THE FORM OF A CONTINUED FRACTKJN. 
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determinants, the following' identical equation : — 
_ /«1«2«3- 
■•am- 
I X 
VajKj®:!-' 
\a1a2. 
• l^m+1 
/ m — l^m \ ^ — l^)n+ l\ 
and consequently 
• •(! m— \ ^ — 1 ^ m'^ m + \ 
Ci jC( 2^3 • • • Oi,n_i • • ’^OT — 1 • ‘ ^m + \ 
' (J iQ-2^Z‘ • ^ /^l^2 * • • l^^m + 1 
dyCi^a^- • ^1*^2 • • • ®^/n — l®^m + 1 
^ U^Qn. . fl'nj 
CJjCJ2”-®^7n-l*®^m+l 
and consequently when 
a^a^...a^_y a„ 
a,c4,...a 
W2 — 1 * ^m + 1 
= 0 
Ci\0-2* • •dfn—X j 1 • + 1 
and 
^ 1^2 • * • 1 ^ 1^2 * ' * ^/n — 1 * 1 
will have different algebraical signs, it being of course understood that all the quantities 
entering into the determinants thus nmhrally represented above are supposed to be 
real quantities. This theorem, translated into the ordinary language of determinants, 
may be stated as follows : — Begin with any square of terms whether symmetrical or 
otherwise, say of r lines and r columns ; let this square be bordered laterally and 
longitudinally by the same r new quantities symmetrically disposed in respect to one 
of the diagonals, the term common to the superadded line and column being filled up 
with any quantity whatever; we thus obtain a square of (r+l) lines and columns; 
let this be again bordered laterally and longitudinally by (r+ 1 ) quantities symme- 
trically disposed above the same diagonal as that last selected, the place in which 
this new line and column meet being also filled up with any arbitrary quantity; and 
proceeding in this manner, let the determinants corresponding to the square matrices 
thus formed be called D,._i, D,., D,.+i, D,.+25 •••• this series of quantities will possess 
the property, that no term in it can vanish without the terms on either side of that 
so vanishing having contrary signs. Thus if we begin with a square consisting ot 
one single term, we may suppose that by accretions formed after the above rule it 
has been developed into the square (M) below written, and which of coarse may be 
