166 PROCEEDINGS OP THE AMEBICAH ACADEMY. 



and (38) becomes 



M(S) ,+ (a>** + iP-*-* "" 



where / is intrinsically positive ; but each of these last integrals has 

 an integrand that must be either zero or positive at every point in its 

 domain, so that u must be independent of a? and y, and must vanish 

 on 8 at every point. It follows that // is everywhere zero and that 

 U' = W. 



It is evident that the condition (3) might have been stated in the 

 form of the equation 



where the integration is to be extended over so much of the plane 



: = : as lies within S IV 



If the space within >% were cut up into portions (filaments) by the 



cylindrical surfaces S u S,, S :h ■ ■ ■, the generating lines of which were 

 parallel to the z axis, and if within each filament 

 Z/(U r ) vanished, while, in addition to the other 

 requirements enumerated above, M'were constrained 

 to have at every point of the surface of every filament 

 the value (TPs), which points with the same z co- 

 ordinate on the surface S„ had, — though the normal 

 derivative of M r at the common surface of two fila 

 l ■'].. i be 58. ments were not expected to be continuous we 

 might assume as before that two dilferent functions 



could satisfy all these conditions and denote their difference by u. 



We could then apply (35) to every filament separately ( Figures 57 



and 58) and obtain from each an equation of the form 



.f^./:/-(:>5)-./:/J[(::)^(:)1— 



(42) 



where li denotes a cross-section of the filament. If, then, all these 

 equations were added together, the re-lilting equation would be 



/-*//(S+Sh-///[ffi) ,+ «)V-* 



which is (35). In this ease also, therefore, M is determined. 







