400 
MESSRS. W. E. AYRTON AND H. KILGOUR ON THE 
cl~t 
^- = p(^-g^(0-Q/(0 
Cl. 
( 9 ) 
where 
P = 
4 x 9-858 
and 
_ 4KA 2 > 4 x 9-858 
u 'ird? 7 rd? 
P and Q are therefore constants for a particular wire with a particular current flowing 
through it. 
We next tried to expand xp (t) and f (t) as functions of t from the curves which we 
had experimentally obtained connecting emissivity with temperature and resistance 
with temperature, but we found that it was not possible to express these functions of 
t in any such simple shape as would allow the integration of equation (9) to be effected 
analytically, and a result obtained suitable to be used for easily determining the value 
of t for any value of x. We, therefore, consulted Professor Henrici regarding the 
integration of equation (9) in a practical form, and we have to express our thanks to 
him for the warm interest that he has taken in this mathematical problem, and for the 
many suggestions that he has kindly made, and which have enabled us to arrive at 
the following solution. We have also to thank one of our assistants, Mr. Walker, 
for carrying out the graphical and numerical calculations contained in this section of 
the paper. 
It is clear that the law of distribution of temperature along the wire will depend on 
the diameter of the wire among other things, also on the current passing through it, 
the variation of temperature with length of wire being the more rapid the thicker the 
wire and the greater the current passing through it. We therefore selected for 
consideration a wire of mean diameter, namely, that of G mils or 0'152 millim., 
and we took the case when L "4 ampere was passing through it, which is the greatest 
current that was passed through this wire in the emissivity experiments. 
Having selected this wire and current, the next step consisted in calculating 
P (t — t 0 ) xfj ( t ) and Q f (t) for different values of t. t 0 is 12° C., P is 2587*4, and the 
value of i/; (t) may be obtained from the emissivity curve for the 6 mils or 0-152 millim. 
wire given in fig. 11. Instead of calculating f (t) the resistance per cubic centimetre 
of the material for different values of ( t ) it is more convenient to write 
Q f(t) as 
KA 2 
~ l 
4 x 9‘858 
7 rd~ 
F(0, 
where l is the length, 17'07 eentims., of this 6-mil wire that was used in the experi¬ 
ments from the results of which the curve in fig. 14 (Plate 15) has been drawn, and 
