and Surfaces of Incandescence Lamps. 373 
and taking logarithms of both, we have the figures given in 
Table I. 
Writing down the differences between successive values of 
log v and log /, we get the numbers— 
Pee ants vas 11621 MVD ie se ain tls 10513 
BEEIUET feiss «now 11009 AO sands ee 11600 
AAG wrccceeee 11323 ADA Cicse ness 10035 
<td capeaeeec 10998 AES. Shes oes 10449 
4:3 Og eens 10619 AUG coaatones 12071 
Now, if /=Av-*, then 
log /—log 4, = —a(log Av—log Av), 
where / and J,, v and v, are adjacent values of J and v; 
accordingly, 
_ logl—log 
log Av—log Av, 
The mean of the differences of log / divided by the mean of 
the differences of logv gives 25:4 nearly; and, accordingly, 
M. Foussat’s numbers agree with those found in America in 
assigning to « a value not far from 25 for the Hdison lamp. 
In the neighbourhood of 100 volts 
[eels ee 
or, average life varies inversely as the 25th power of electro- 
motive force. 
This, however, is easily seen to be a very rough approxi- 
mation. ‘The successive quotients of life-difference by volt- 
difference are not constant; and the above simple exponential 
formula cannot be admitted as anything more than a very 
imperfect connection. In examining the Carlisle Tables of 
Mortality, which give the expectation of life at every age 
drawn from a very large number of observations, it was appa- 
rent that an empirical formula connecting the two quantities 
could be obtained of a form 
log e=a+be-+ cx’ + &e.*, 
where e=expectation of life at any age a. 
I was led therefore to try if such an empirical formula 
better suited the present case; and a process of trial and 
* Let z be the age, and e the expectation of life at that age. Then by 
the Carlisle Tables of Mortality at the several ages 10, 20, 30, 40, 50, 60, 
70, 80, 90, the corresponding expectations of life are :—48°82, 41°46, 34:34, 
27°61, 21:11, 14°34, 9°18, 5°51, 8:28; and it can easily be found that 
x 14 
spe iyo alae Seley ai. 
sole ee oO 10000 
the calculated values from which are :—49'6, 44, 36-6, 28°6, 20°9, 14:3, 9-64, 
5:54, 3°14. 
x*, very nearly ; 
