266 Proceedings of Royal Society of Edinburgh. [ sess . 
The supposition therefore that the equation of the n th degree S = 0 
can have a pair of imaginary roots leads us to assert that a perfectly 
similar equation, P a . x = 0 , of the (n - l) th degree, will have the 
same pair of roots. It is thus seen that the supposition and 
reasoning, if persevered in, will ultimately land us in an absurdity, 
when we reach, as we are bound to do, one of the equations of the 
first degree 
Rxx ^ = 0 5 Ryy ® = 0 , ’ 
“ Done l’equation S = 0 n’a pas de racines imaginaires.” 
The next object being to fix the limits between which the roots 
of the equation S = 0 are comprised, a theorem necessary for the 
accomplishment of this is first attended to Formally enunciated 
in modern phraseology it is : — 
S being any axisymmetric determinant , E the determinant got by 
deleting the first row and first column of S, Y the determinant got 
by deleting the first row and second column of S, and Q the deter- 
minant got from E as E from S, then , if E = 0 , 
SQ = -Y 2 . 
As the mode of proof employed by Cauchy applies equally well 
when S is not axisymmetric, let us take | a 1 b 2 c 2 d 4: \ for the given 
determinant, and write the proof as it would nowadays be given. 
To begin with, if A 1 , A 2 , ... be the complementary minors of the 
elements nq , a 2 , ... in | afi.f % d | we have 
a l A 1 - a 2 A 2 + a s A 3 - a 4 A 4 = \afiffif, j 
&iA 4 — b 2 A 2 + &3A3 — 6 4 A 4 = 0, 
^iAj c 2 R 2 + C3A3 c 4 A 4 0 , 
d^ A 4 — d 2 A 2 + 6? 3 A 3 — d^ A 4 = 0 . J 
Flitting A 4 = 0 , and leaving out one of the last three equations, we- 
obtain 
— a 2 A 2 + oqAg — a 4 A 4 = | af) 2 c z d± | , 'j 
“ c 2 A . 2 "H C3A3 c 4 A 4 0 , 
- d 2 A 2 + d B A 3 - d± A 4 = 0 j 
from which by solving for A 2 there results 
A _ ~ 1 uf) 2 cfi^ | • [ cfi 4 
2 ~ I I 
