406 
MR. 0. H. LEES ON THE THERMAL CONDUCTIVITIES 
v 1 — A cosh ctXy + B sinh ax l | 
v. 2 = A cosh ax 2 + B sinh ax 2 V 
r 3 = A cosh a.r 3 -j- B sinh a.r 3 
These three equations are sufficient to fix the values of the constants A, B, a, but 
their solution is difficult unless x 3 — x % — x % — x v This relation is very nearly 
satisfied by the points of observation, in the hot bar, x 2 — x l = 10*56 cms. x 3 — x 2 
= 10*44 cms. ; in the cool bar, x 2 — x 1 = 10*51 cms. x s — x 2 = 10*48 cms. 
The error introduced by assuming each of these intervals =10*5 cms. is small 
enough to be neglected in the cool bar, but a small correction is necessary in the 
case of the hot bar. 
If dvjdx be the value of dv/dx at the point x 2 we have for the temperature v 2 at 
the point x 1 -f* 10*5, 
vd = Vo — *06 ~ . 
ax 
Now dv„Jdx is found to differ little from 1 for any of the experiments, 4 * and we 
may, therefore, with sufficient accuracy, take v 2 = r 2 — *06. 
Making use of v 2 we have from the three temperatures, v 1} v 2 , v 3 
cosh od = ■ , 
where l = x 2 — x y — x 3 — x 2 , &c. This equation determines a from any three obser¬ 
vations of temperature. The mean of the values thus determined for different points 
along the bars and for different experiments is used in the subsequent work. 
We have then the equations 
v l = A cosh ax y -fi B sinh ax y 
V 2 = A cosh ax 2 + B sinh ax 2 
* The value of dv^dx may be determined as follows. By Tailor’s Theoi’em we have— 
/ o + o -/(») = if oo + |r (*) +|r (») + &c. 
/ (a;) - / (a -/) = If (x) - If O) + If" (x) + Ac. 
Therefore 
f(X+T) 2 , f(X -- =/ 00 + | f" («0- 
Putting v —f (re) we have, since for any short length of the bar = a~v, 
f" (re) = (®). 
/» = /J;. + 0-/(y . -0 , 
2! ( 1 + wj 
Tlie value of i(Z-ar) for this-point of the bar is in nil the experiments approximately *04. 
Therefore 
