516 Dr. 0. J. Lodge on the Variation of 



For a much simpler and practically useful form of this equa- 

 tion see equation (40) § 38. 



36. The expression on the left-hand side of this equation 

 may be written in various ways ; and its form is slightly differ- 

 ent according as C is of the same or opposite sign to k. 



For the case when C and k have the same sign (which ac- 

 cording to Prof. Tait is probably most usual) we may write 

 it conveniently 



— 



T ,(«\ • k-i ~ e ( K \ • u-i 2C0 + B y 



[w( x )smh ' ^^^ -^ 1 7(4AC^^)Je = 



y/(R<r*)a:. . . (36) 



But for the case when the signs of C and k are opposite (as 

 for iron), it becomes 



2 A+n 



KAj COsh 7TB^4AC7~ V (Tr) C0S \/(B^-4AC) Je = 



>/(R«7«>. . . (37) 



Whether we write cos -1 or sin -1 in this equation only affects 

 the sign of the term containing it. In equation (36), if 4AC 

 is less than B 2 (which is unlikely), the terms in the denomi- 

 nators must be transposed and cosh -1 written for sinh -1 . 



Of the two terms in the brackets of these equations the first 

 is by far the most important, and in the former paper is the 

 only one which appeared (see equation 14) ; the second term 

 has only a small effect on the curve, and this effect vanishes 

 with C*. The occurrence of the inverse circular function in 

 the curve of temperature along metals whose conductivity di- 

 minishes with temperature is peculiar ; but C must always be 

 very small for such metals ; so that it does not make much dif- 

 ference. And this is a good thing, because when m is nega- 

 tive the 6 occurring in the expression called C is of more rela- 

 tive importance, and C is therefore not so constant as when m 

 is positive : but the approximation is always pretty good, the 

 worst possible case being that of a supposititious metal with 



m=: — ln ), for which C = 1 + tt^' 



v a/ ; 3oo 



37. We have thus obtained the equation to the curve of 

 temperature expressing x as a function of 6. What we have 

 now to do is to show how, from experimentally observed cor- 



* To avoid a possible misunderstanding, it may be well to say that this 

 does not mean that the term vanishes, because of course it becomes infi- 

 nite ; but it means that the effect of the term vanishes, because when the 

 limits are put in, the two things subtracted from one another are equal. 



