664 Proceedings of Royal Society of Edinburgh. [sess. 
the paper from our present point of view is the fact that P and R, 
are changed so as to have the zero elements on the under side of 
the main diagonal matched by like elements in the corresponding 
places on the upper side, with a view to expressing the root of the 
equation in the form of an interminate continued fraction. The 
procedure need not be described at length ; it will be understood 
at once by applying it in the case of the determinant 
a l a 2 a 3 1 
CL~y $2 $3 
% a 1 a 2 
a o ? 
— -a course which also enables us to test the accuracy of the illus- 
trative example given later in the paper. In the first place, the 
element in the place (1, 4) is changed, the result being 
a 2 
a 3 
a i 
a 2 
af 
- 
a o 
a 1 
a 3 a 2 
- a 
a 0 
d^CL-^ 
-a. 
secondly, the element in the place (1, 3) is changed, the result 
being 
a 2 
a 0 
a i 
af 
CL^Ci^ 
a Q 
a 2 a Y 
- a 3 a { 
a 2 a 0 
Ct-^ 
CIqCL^ CSq 
. a 3 a 2 ; 
and lastly, the element in the place (2, 4), with the result 
a x a 2 
a 0 a 1 af - a 3 a Y 
a 0 a 2 a Y - a 3 a 0 ( a 2 2 - a 3 a^) (a 3 a 2 — cq) - (a 2 a i ~ a 3 a o) ( a 2 ~ a %) 
a 2 a Q ( a 2 2 — ay^) (a 3 6q — a 0 ) - a 2 afaf - a 2 ) 
a 3 a 2 ( a 2 2 - a 3 oq) 
It is thus seen that if the equation for solution be the quartic 
a* 4 + a 3 x 3 + a 2 x 2 + a Y x + a 0 = 0 , 
a convergent to the smallest root is, according to Piirstenau, 
