On the Influence of Temperature of Liquid Air on Bacteria. 339 



has for indefinitely increasing values of n, as limit p, tends multiplied 

 by (1 - towards the limit /) r(X +1). This proposition, which 

 follows immediately from the type theorem, gives for X = Abel's 

 theorem; for X = 1, the theorem of Frobenius (' Crelle,' 1878); and 

 leads to consequences in regard to the integrals of linear differential 

 equations with rational coefficients. The principal result of this 

 chapter, is that the singularities of a function, defined by a power 

 series Sc^a;'^ can be found and their nature analysed by the examina- 

 tion only, for values of n tending towards oo , of the sequence Cn and 

 of other sequences derived from this one. 



A new conception is introduced in the third chapter, that of an 

 " asymptotical region." An asymptotical region encloses always a 

 point X = a, of essential singularity, of a function F(a;) and consists of 

 an ordinary region enclosing a with an infinity of co-holes in it, not 

 enclosing ft, but approaching it indefinitely. The object of the author 

 is to throw some light on the manner in which a function behaves in 

 the vicinity of a point of essential singularity. It is shown that if 



a being oo , and the mn denoting finite integers, an asymptotical region 

 can generally be constructed in which lim. '¥{x) = if lim. a; = oo ; and 

 that this proposition is convertible. The new conception is applied to the 

 theory of transcendental integral functions as founded by Weierstrass, 

 Laguerre, Poincare, &c. If G(ic) denotes a function of class zero, a 

 certain asymptotical region will belong to G'/Gr in the above sense. If 



H(a') = (j{x) + Ci (ji\x) + C.2 G:"{x) where the ci are any con- 



2 c-t^ 



stants, such that the power series , . , * , has a non- vanishing 



=1, go) 



radius of convergence, YL{x) will again be a transcendental integral 

 function of class zero ; and the asymptotical region belonging to H(x) 

 will be the same as that belonging to G(a:). 



Further ]^ote on the Influence of the Temperature of Liquid Air 

 on Bacteria." By Allan Macfadyen, M.D., and S. Eowland, 

 M.A. Communicated by Lord Lister, P.RS. Eeceived 

 April 3,— Eead April 5, 1900. 



In a previous communication"^ it was shown that no appreciable 

 influence was exerted upon the vital properties of bacteria when ex- 

 posed for 20 hours to the temperature of liquid air(-183°C. to 



* ' Roy. Soc. Proc.,' February 1, 1900. 

 VOL. LXVI. 2 D 



