The theory of determinants in the historical order of development, by Sir Thomas Muir.

CONTINUANTS (SCHLAFLI, 1858)

437

which for shortness' sake he denotes by
a (a, /,,y...,,0)
and whose connection with continued fractions he therefore
specifies by the equation
A(a,/3,,...,,) = 1 c cos2p
cos2?
1 -cos20
The first property noticed is, naturally,
A (a,/,,,... 0) = A(/,,...,0)- cos2 ~ A(y,...,).
Later there is given what may be viewed as an extension of this,
viz.,
A(a,...,.,,,, 0,...,X) = A(a,...,,e). A( 0,..., A)
-COS2. A(a,..., d). (0,..., ),
the proof being said to present no difficulty. The third is a little
more complicated, and is logically led up to by taking four
instances of the first property, namely,
A(a, 23,y,.., A) = A(/3,y,..,, - cos2azA(y,, 6,..,),
(y,, 5,...,, ) = A( (,,.,,) - cos2/3 A(,.., ~, ),
A(y-, 8,...,,, 0) = A(y,, 8,.., 1) - cos20.AZ(y,,...,),
A(...,...,, 0, ) = A(s,...,,, 0) - cos2a (8,...,, ),
using in connection with these the multipliers
A(3,...,,~), - A(~,..., ), A(~,...,0), -A(7,5,..., ),
respectively, performing addition, and then showing that the
right-hand sum vanishes, the result thus being
A(a,B3,y,,,...,,)A(,)-A(,..., C,0,a). (7,,..., A)
- {A(3,B,.,. *., ~,1) - A(,, **..., ~,i,0)}.A (,..*.., ).
The fourth property concerns the determinant
A(/,y,...,,0) A(a,/,y,... I,0)
A(/,y,...,) A(a,A,3, y,, ) 1,