Geological Society. 233 



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

 circuit. Then 



k—l-=2m + m' — {2n + n'). 



Sturm's demonstration of the case where m'=0, w=0, m'=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—l 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 mk—l 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. 



GEOLOGICAL SOCIETY. 



[Continued from p. 76.] 

 May 26, 1858.— Prof. Phillips, President, in the Chair. 



The following communications were read : — 



1 " On the Pleistocene Sea-bed of the Sussex Coast, being the 

 Western Extension of the Raised Sea-beach of Brighton." By 

 J. Prestwich, Esq., F.R.S., F.G.S. • j , i. 



The author first observed that the well-known raised beach 

 at Brighton described by Dr. Mantell, has been since extended by 

 Sir R Murchison to Hove, and by Mr. Dixon to Lancing and 

 Broadwater near Worthing; and that Mr. Godwin-Austen has 

 d-scribed some marine beds, which he ascribes to the same age, on 

 the coast between Bognor and Bracklesham. He then proceeded 

 to state the results of his examination of the country between 

 Brighton and Havant. Traces of this old sea-beach are, in Mr. 

 Prestwich's opinion, to be seen on the slightly-raised ground at 

 the base of the chalk-hills east of Arundel, at a distance of three 

 or four miles from the sea. It is much more distinct westward of 

 Arundel, and can be followed to near Chichester. In a wood at a 

 short distance north of the road from Arundel to Chichester, and 

 about two miles from the former, these beach-deposits appear as a 

 bed of sand about seven feet thick, with thin patches of shingle, and 

 overlaid by subangular flint-gravel. On the brow ot Avisford HiU 

 the same bed of sand is seen to overlie the mottled clays ; and m 

 a sand-i)it in the wood cast of Slindon Common, simdar pebbly 

 sand at least 16 feet thick, is seen to be overlaid by partially 

 rounded flint-shingle, about three feet thick, and both to be covered 

 by ferruginous sandy clay full of large angular flints, winch occu- 

 pies also a furrow excavated through the shingle into the sand. 



