475 



platinum and iron. To prevent the apparatus from acquiring 

 in time the same temperature, the alternate junctions of the 

 metals, to which the heat is not applied, might be con- 

 nected with the pedestal upon which the lamp is placed ; 

 and the pedestal be either allowed to rest in a cold situa- 

 tion, or else be connected by wires with some extensive 

 cooling surface." 



The President gave an account of some investigations re- 

 specting Fluctuating Functions, from which the following 

 are extracts : — 



" Let t? x denote any real function x, continuous or discon- 

 tinuous, but such that its first and second integrals, 



\ dx p x , and A dx\ p x , 



are always comprised between given finite limits. Let also 

 the equation 



(j o / ^ — f 1 ' 



in which fx is some given constant, have infinitely many real 

 roots, both positive and negative, which are not themselves 

 comprised between any finite limits, but are such that the 

 interval between any one and the next greater root is never 

 greater than some given finite interval. Then, 



t =» J/^J d V*» F * = °» ( A ) 



if a and b are any finite values of x, between which the 

 function y. v is finite. 



"Again, the same things being supposed, let the arbitrary 

 function F* vary gradually between the same values of x ; 

 and let v x be finite and vary gradually when x is infinitely 

 small; then 



F ff ziar- 1 J dt J a dx Ptx-ty *■& (#<&)> (B) 



2s 



