or 
ary 
OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. ve 
VIII. The investigation shows that, for example, S, is proportional to 
(1234)a’ + (1235)a’ + (1236)z + (1237) (38), 
where (1235) is the determinant formed from the Ist, 2d, 3d, and 5th columns 
of the matrix, 
5; 5¢, 100; 106, Sf, ¢,. . 
@,.00,, 10e , 10d =-45e PUP; 
-» U, Se, 10d, 106°, bf, g 
. , (ey bbi, Oct 1d »,. 5e— f 
and in like manner for (1234), &c. :—the remaining factor in S,, being inde- 
pendent of z. In other words, S, is proportional to the determinant, 
: : ; ; ai aia ge atl d (39). 
’ ’ ? ) 1 ae es 
CI 1) ee) ? 1 es . ? 
b206. piOd,, IOerewbe Big: 
@,005, 80c. Od oe: fF 
By pac. 10d 10e.) 5k og 
a ok DUN, LOG. 10d joe. OF 
Note.—The signs of the constituents in the last four lines of the 2d, 4th, and 6th columns of this 
determinant are minus (see e.g., (7) or (8) ; but attention has been paid to this in equations (40). This 
has been done for simplicity of writing. 
This determinant is equal to another formed from it in the following manner :— 
Let C, , e.g., represent the 7th column ; write. 
instead of C, , C,a°+ C,a°+ Cya*+ Cyr? + Ca? + Cr+, | 
instead of C, , 6C,a°+ 5C,2*+ 4C2° + 8C,2° + 2C,0 +C, | 
instead of C, , 15C,2*+10C,2° + 6C,2° + 3C,2 + C, tn (40) 
instead of C, , 20C,2°+10C,2°+4C,2 + C, : 
instead of C, , 15C,2’°+ 5C,x + C, | 
instead of C, , 6C,a7+ C, J 
and leave the first column as it is. 
This determinant will be found to reduce at once, to one of the fourth order, 
by the loss of its 1st, 2d, and 3d lines, and its 4th, 5th, and 6th columns; and 
thus 8; is proportional to 
VOL, XXIX. PART II. 6Z 
