1894.] Mr Dixon, Geometrical proof of Gonvergency. 217 



without prejudice to any theory as to the actual cause of the 

 phenomena. 



(4) On a new method of preparing culture media. By Dr 

 LoRRAiN Smith. 



The author described a method for preparing media suitable 

 for the cultivation of Bacteria. The principle of the method 

 consists in the addition of a small percentage of alkali to fluids 

 which contain proteid such as egg-white and serum of blood. 

 The fluid is then heated to the boiling point or over it in the 

 autoclave. By this means it is converted into a clear transparent 

 jelly. It is then a medium suitable for the growth of a large 

 variety of germs. 



Monday, May 28, 1894. 

 The following Communications were made to the Society : 



(1) Exhibition of nest of Trochosa picta and of certain well- 

 marked varieties of this spider. By C. Warburton, M.A., Christ's 

 College. 



(2) Exhibition of Magnetic Rocks. By Mr S. Skinner. 



(3) Geometrical proof of a Theorem of Gonvergency. By 

 Mr A. C. Dixon, Trinity College. 



Let Wj + ^2 + Uz + ... be a convergent or oscillating series of 

 quantities which may be complex, and a^, a.^, a^ ... a series of real 

 positive quantities diminishing continually and without limit. 

 Then the series tti^i -}- ag'^'a + ^3% + • • • will converge. {Ghrystal, 

 Algebra, Chap, xxvi, § 9, Theorem IV.) 



Let us write 



2«n for lii+lh+ ••• + Un and SrUn for Ur + Ur+i + ... +Un. 



Since the series U1+U2+ ... does not diverge, a circle of finite 

 radius (R) can be drawn on Argand's Diagram which will include 

 the point Xun for every value of n. 



The point ttiXun will therefore always lie within a certain 

 circle of radius a^R, which may be derived from the former by 

 drawing lines from the origin to its circumference and multiplying 

 each by a^. If now the origin is moved to the point aiiii, the 

 quantity aiU^ is subtracted from every quantity represented, and 

 the point a^iX^Un therefore always lies within the circle as con- 

 sidered from the new origin. Also the new origin is inside this 

 circle, for it was the point ai%Ui. 



The point aS,^Un will therefore lie within a circle of radius a^R 

 formed by diminishing every line drawn from the new origin to 



VOL. VIII. PT. HI. 16 



