Conductivity of Metals with Temperature. 253 



we get the radiation part of (12) in the form 



where a means log a, and where 7 is written for the correc- 



1 © 



tion factor ~ a ©, or inr>n * And this equation simplifies to 



(^y = E«^{m + g(2 + ^a + 7 (^-l))+^(l+7)}.(13) 



Although, therefore, the equation looked as though it would 

 contain, when put into the ordinary form for integration, a 

 quartic expression under the root, and therefore would land 

 us in elliptic functions, yet on working it out we find that 2 

 is a factor of the expression ; and hence, the expression under 

 the root being only of the second degree, the integration can 

 be performed without difficulty. Integrating between the 

 limits and m, and between ® and 6, and remembering that 



— is essentially negative, we get as the result, 



-7T + 2 + ma. + ry(ma. — 1) 



sinh- , 6 ! = WKm«, . (14) 



L . vf 12m*(l + y) — {2+7wa + 7(ma — l)} 2 Je 



or 



sinh" 1 ^ + j) = sinli- 1 ^g-+Jj+a? v /R w « ; . . (15) 



where L and J are used as abbreviations for certain evident 

 expressions such that the ratio of J to L, which is all we need 

 trouble about, is 



J,2- y Wl + y) = XMy (16) 



It is quite possible, however, for L and J to be both imaginary, 

 which happens when the second term under the root on the left- 

 hand side of (14) is greater thau the first. When this is the case 

 I suppose the sinh becomes cosh, passing into it through infinity ; 

 but the ratio J : L, or A, is always real and positive. 



Beferrring back to § 7 and to equation (9), we see that a likely 

 value of m for iron is 300, and for copper 650 ; so that for these 

 two metals the value of ma is about 2*3 and 4*9 respectively. 

 Looking at the expression under the square root on the left-hand 

 side of (14), and neglecting squares of the small quantity y, one 



