Influence of Bending of Needles on Magnetic Dip. 275 



which completes the solution, and leads to results similar to 

 those of the latter portion of Section 6, e. g., that surfaces 

 equipotential initially are also so finally, and the surface of 

 maximum temperature has the same potential, the mean of 

 the electrodes, initially and finally. 



Similarly we may extend the corresponding results of 

 Section 7. 



From the maximum theorems which have been obtained, a 

 general result as to the superior limit of the temperature in 

 any part of a conductor is easily derived. It is only necessary 

 to notice that any further escape of heat than is allowed for in 

 these theorems cannot raise the temperature at any point, but 

 must, on the contrary, lower it more or less ; hence : — 



If in any conductor heated by a current a closed surface, 

 or two surfaces which, by cutting through the boundary, 

 enclose a portion of the body between them, be taken, then the 

 highest temperature in the enclosed portion cannot be greater 

 than the highest temperature on the closed surface or on the 

 cross-sectional surfaces — provided, in this latter case, no heat 

 flows into the enclosed portion through the remainder of the 

 boundary — by more than, though it may be as much as, the 

 value §', which is given by /{$') =jW/S, where E is the 

 greatest difference of potential between any two points of the 

 closed surface or pair of surfaces, and/'($) is the ratio of the 

 thermal and electric conductivities. 



By " any conductor " is, of course, to be understood, in the 

 general case of heterogeneity, one with conductivities subject 

 to the limitations introduced in this Section. 



A similar result in the case of two dimensions may be at 

 once derived from the maximum theorem of Section 7, but it 

 is unnecessary to state it. 



on 



XXX. The Influence of the Bending of Magnetic Needles 

 the apparent Magnetic Dip. By Arthur Schuster, F.R.S. 



IN Maxwell's Treatise on ' Electricity and Magnetism/ 

 vol. ii. page 117, the following passage occurs: — 

 " Dr. Joule finds that the needle should not be more than 

 five inches long. When it is eight inches long, the bending 

 of the needle tends to diminish the apparent dip by a fraction 

 of a minute." 



I have not been able to find in any of Joule's writings a 

 passage on which the above statement could be founded. Such 

 a passage may have escaped me ; but it is not impossible that 



* Communicated by the Author. 



