UNDER THE FORM OF A CONTINUED FRACTION. 
425 
{p, when even, as 2p, with a square of 4 terms, 
{{p—‘2), (p-])), {(p~2),p) 
{(p~2), p), ((p~l),p). 
Each stage may be considered as consisting of three parts, a diagonal set of equal 
terms transverse to the axis of symmetry, and two triangular wings, one to the left, 
and the other to the right of this diagonal ; the terms in each such diagonal for the 
respective stages will be 
(0,71); (1,72—1); ( 2 , («-2)) ; . . (/?, (/?+!)), 
p being 1 when 71 is even, and when n is odd. 
If we change the order of the coefficients in each of the two given functions, it will 
be seen that the only effect will be to make the left and right triangular wings to 
change places, the diagonals in each stage remaining unaltered. The mode of 
forming these triangles is an operation of the most simple and mechanical nature, 
too obvious to need to be further insisted on here. 
Art. (8.). When we are dealing with two functions of unequal degrees, w and n-\-e, 
we can still form a square matrix with the coefficients of the two systems of (e) and 
(n) equations respectively, but this will no longer be symmetrical about a diagonal ; 
it is obvious, however, that if we treat the function of the lower degree, as if it were of 
the same degree as the other function, which we may do by filling up the vacant 
places with terms affected with zero coefficients, the symmetry will be recovered ; 
and it is somewhat important (as will appear hereafter) to compare the values of the 
Bezontian secondaries as obtained, first in their simplest form by treating each of the 
two functions as complete in itself, and secondly, as they come out, when that of the 
fniictions, which is of the lower degree, is looked upon as a defective form of a 
function of the same degree as the other. A single example will suffice to make the 
nature of the relation between the two sets of results apparent. 
Take fx—a x‘^-\-cx^-{-dx-\-e 
(px=0 .x‘^-\-0 .x^-^-yx^-^r^x-^s. 
The general method of art. (7.) then gives for the Bezoutian matrix 
0 ay ah as 
/ ad as 
ay 
ah 
as 
hz 
\ 
cz — ey 
hz 
^ch — dyj 
cs — ey dz — eS. 
We shall not affect the value either of the complete determinant, or of any of the 
minor determinants appertaining to the above matrix, by subtracting the second line 
of terms, each increased in the ratio of & : a from the first line of terms respectively ; 
3 K 2 
