On the Heating of Conductors by Electric Currents. 259 



portional to the size of the object. For larger angles the 

 proportionality may still be made to hold by a slight alteration 

 in the focal adjustment ; and to this degree of approximation 

 we have 



[ \ <p(x) cos hcdx]*+ [ \ c/>(V) sin kxdx~\* 



Tj>(^P~ "' ' ' 



If the source is symmetrical the second term vanishes, and 

 the expression reduces to the original form. 



It is possible that, in addition to the uses already mentioned, 

 the " visibility-curve " may have an important application in 

 the case of small spherical nebulae. For from the form of 

 this curve the distribution of luminous intensity in the 

 globular mass may be inferred, which would furnish a valu- 

 able clue to the distribution of temperature and density in 

 gaseous nebulae. 



When the source is so small as to be indistinguishable from 

 a star, it would seem that this method is the only one capable 

 of giving reliable information ; but even in the case of bodies 

 of larger apparent size it is equally applicable, may be made 

 to give results at least as accurate as could be obtained by 

 photometric measurements, and is far more readily applied. 



XXIX. On the Heating of Conductors by Electric Currents, and 

 on the Electric Distribution in Conductors so heated. By 

 J. McCowak, M.A., B.Sc, Assistant Lecturer on Natural 

 Philosophy, University College, Dundee*. 



SO far as I am aware, no problems in thermal distribution 

 involving an internal generation of heat — such, for 

 instance, as takes place in the passage of an electric current 

 through a conductor — have hitherto been discussed. Fourier, 

 the founder of the mathematical theory of thermal conduction, 

 takes no account of possible internal sources of heat in forming 

 his equations, and considers only problems in which the sources 

 are superficial. The same may be said of Poisson, Lame, 

 Riemann, and others who have since applied and extended 

 Fourier's methods. Thomson, in his synopsis of the mathe- 

 matical theory f, has expounded Fourier's diffusion-analysis 

 in a form directly applicable to cases where there are internal 



* Communicated by the Author. Read before the Edinburgh Mathe- 

 matical Society, January 9th, 1891. 



t Appendix to article " Heat," Encyc. Brit. 9th edition. 



