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 p, denote any real function x, continuous or discon- 
tinuous, but such that its first and second integrals, 
‘ dz P,, and (‘’, dv) Ps 
are always comprised between given finite limits. Let also 
the equation 
(V, dz) ; Bass fs 
in which yu 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, 
ats : dx . dyPy Fz= 0, (A) 
if a and b are any finite values of x, between which the 
function F, is finite. 
« Again, the same things being supposed, let the arbitrary 
- function Fz vary gradually between the same values of x; 
and let pz be finite and vary gradually when z is infinitely 
small; then 
=o \" dt ‘. dx Prz—ty Fxy (y25)> (B) 
2s 
