The leads of a tungsten filament in vacuum cool the ends of the filament and so affect the voltage, candle power, electron emission and other properties of the filament. For long filaments, where there is a central portion at a uniform temperature __\( T_m \)__, the temperature distribution near the lead is derived. A method for determining __\( T_0 \)__, the temperature of the lead-filament junction, is given. Tables and formulas are presented which allow ready calculation of the effect of the leads on the properties of any long tungsten filament for which the current and diameter are known. From the more general results it has been found that the decrease in voltage due to the cooling of one lead may be represented by
__\[ \Delta V=0.154\frac{T_m}{1000}-0.081\frac{T_0}{1000}-2.1\cdot 10^{-8}T_0T_m-0.056 .\]__
There is an extension of the theory to cover the cases of filaments in gases, filaments of other materials, etc.

Part II of the paper gives figures from which may be found the properties of filaments so short that the first theory does not apply. Some experimental checks of the theory are given.

In general the results and the methods of application have been placed first, and the mathematical derivations have been placed at the end of each part.

For a short filament with leads cooled in liquid air a negative slope of the voltampere characteristic when the central temperature is much smaller than __\( T_m \)__ is observed.