258 Dr. 0. J. Lodge on the Variation of the Thermal 



To find fM from (26), we may either write it thus, 



//|== cosh " 1 ?^ \og p (r + \/r 2 — 1), . . (26') 



or, what is usually simpler in practice, we may notice that for 

 the majority of metals r is but a fraction over unity, and hence 

 that the number /x£, whose hyperbolic cosine is equal to r, 

 must itself be very small, and high powers in its expansion may 

 be neglected with impunity. Writing (26) therefore thus, 



,= eosh # ?= 1 + >*f + 0¥ + g + . . ., 



the last term written is nearly always utterly negligible, and 

 the last but one is usually exceedingly small. Hence a first 

 approximation to pu is 



M?? *2(r-l); (28) 



and a second and generally sufficient approximation is 



y?f ^ 2(v/3To>-3); . . . (29) 



and as £, the distance between successive thermometers, is 

 known, p, is determined. 



Expression for the Relative Conductivities of two Metals in 

 terms of the Constant r. 



26. Equation (22), for the relative conductivities of two 

 metals under precisely similar circumstances, becomes, there- 

 fore, if the intervals f are the same for both, 



K = g 0?0 /l 0g(/ + V / /* II l) l 2 / 3Q N 



fc'n g'o'X w^j- *A 2 __n J ' v 



V 



P o I log (r + \/r 2 — 1) 



or, to an approximation sufficient for all but very badly con- 

 ducting metals like bismuth, 



h^ c oPo , x/(3 + 6/)-3 ^ c p y—i , n 



where it will be remembered that c and p mean respectively 

 the specific heat and the density of the metal at zero Centi- 

 grade. 



To determine the absolute conductivity, we must determine 

 the constant P by direct experiments on cooling (see equa- 

 tion 23). 



On the Determination of the Radiation- Constant P. 



27. The red, or a bit of the rod, is to be heated, as a whole, 

 to some moderately high temperature, and then placed in the 

 exhausted receiver under precisely the same external condi- 

 tions as it is exposed to during the conduction experiments, its 



