OF CRYSTALS AND OTHER BAD CONDUCTORS. 
491 
-F 20 per cent, in conductivity for 100° C. change of temperature. Taking therefore 
k 0 as equal to &(1 + av) the above equation becomes 
dh 
(1 + av) — 4- a 
do\~ ph 
hi 
= §Ui + H 
Multiplying through by 2 (1 + av ) ( dvjclx ), this becomes 
d 2 v dv 
do\ z dv 
ph 
do 
2 d + <*>?■& s + 2«(» + av ) U) i, = 2 ik U 1 + Hd + <«) i, 
or, 
dv 
d /-—- dv\° ^ ph , , , dv 
i + av di) + « + + 
From which by integration we obtain 
(5) 
1 -j- av c ~) = ~ (v 3 + 2 ' v s + -y 4 ) + constant. 
dxj qk \ o 1: ' 
If the value of dvjdx when v — 0 be dvjdx, the above equation becomes 
dvV 
1 + av dx) - 
do. 
h>h c, ( « + & . cib 0 . 
= # ,r ( l + s 'T ^+2 "*)• 
( 6 ) 
If dvjdx = 0, the integral of this equation can be expressed in terms of logarithmic 
and circular functions. If clvjdx 4= 0, the integration introduces circular functions 
and elliptic integrals of the 1st and 3rd kind. If the problem under discussion were 
—given the conductivities—to determine the distribution of temperature throughout 
the bar, this integration would be necessary, but as we are given the distribution and 
have to find the conductivity, the problem can be solved without further integration. 
Although this method is probably not as accurate as that depending on the integrated 
equation, its accuracy is sufficient for the present purpose, where the value of the 
conductivity is only required over a small range (25°-37° C.) of temperature. 
To determine dvjdx at the points of observation v is represented by an empirical 
function of x and the differential coefficient with respect to x taken. The known 
conditions which v satisfies, fix, to some extent, the function to be used ; we see, 
e.g., that it vanishes for some value of x, and that at that point the first differential 
coefficient is finite ; that it increases in one direction with x in an approximately 
exponential manner. 
These considerations lead at once to the function A sinh (ax -f /3), and this function 
has been used. It evidently cannot express v accurately throughout the bar, since it 
is the solution of a linear differential equation, but it can be made by a proper choice 
of the constants to represent the main feature of the curve, the differences between 
its values and those observed being afterwards represented by an additional expression, 
which has generally only small values. 
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