194 



If pairs of tangents are drawn externally to each couple of three 

 unequal circles, the three intersections of the tangents of each pair 

 will be in one straight line. 



Also a paper was read by Professor De Morgan, " On a Proof 

 of the existence of a Root in every Algebraic Equation : with an 

 examination and extension of Cauchy's Theorem of Imaginary 

 Roots ; and remarks on the proofs of the existence of Roots given 

 by Argand and by Mourey." 



The extension of Cauchy's theorem is very easily found, when the 

 proof is the first of those given by Sturm in Liouville's Journal. 

 The extended theorem is as follows : — 



Let <f>z be any function of z, and let z=x + yV — 1. Let (x, y) 

 be a point on any circuit which does not cut itself. Let this point 

 describe the circuit in the positive direction of revolution ; and, 



p 

 <j>(x + y*/ — 1) being P+Q. V — 1, let — - change sign k times as in 



+ — , and / times as in — 0+. Let (x, y) be called a radical 

 point when <p(x+yV — 1) =0, or=oo. Let there be m radical 

 points of the first kind within, and m! upon, the circuit : let there 

 be n radical points of the second kind within, and n! upon, the 

 circuit. Then 



k — l=2m + m' — (2n + n'). 



Sturm's demonstration of the case where m' = 0, n=0, w'=0, 

 which is Cauchy's theorem, assumes the existence of the roots of 

 an algebraical expression. Mr. De Morgan's proof of the existence 

 of these roots is as follows : — He shows, a priori, that in the se- 

 quence of signs which Cauchy's theorem requires to be examined, 

 k — / never undergoes any alteration except after and oo have 

 coincided, that is, where P = 0, Q = 0, simultaneously. It is 

 then easily proved that change in k — / happens in every algebraical 

 equation. 



The proofs given by Argand and Mourey were intended as illus- 

 trations of the power of the extension which is now called double 

 algebra. Stript of this interpretation, they are purely algebraical, 

 and Argand's proof is really that which was afterwards found by 

 Cauchy. Argand's proof is more simple in form than Cauchy's. 



February 8, 1858. 



A paper was read by the Rev. O. Fisher, " On the probable 

 origin of numerous Deep Pits on some Heaths in Dorsetshire." 



Also a paper was read by Professor De Morgan, " On the 

 Syllogism, No. III., and on Logic in general." 



This paper is divided into two sections, the first of which is de- 

 scriptive and controversial, the second is an abstract of the system. 



