166 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



and (38) becomes 



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 ii must be independent of ,r and y, and must vanish 

 on *S^o at every point. It follows that u is everywhere zero and that 



Tr= 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 i>lane 

 c = c as lies within /S'q. 



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

 cylindrical surfaces Si, S^, S3, ■ ■ ■ , the generating lines of which were 

 parallel to the ;:; axis, and if within each filament 

 L (TI'} vanished, while, in addition to the other 

 requirements enumerated above, W were constrained 

 to have at every point of the surface of every filament 

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

 C) n 0> ordinate on the surface >% had, — though the normal 



1 derivative of W at the common surface of two fila- 



FiGURE 58. ments were not expected to be continuous, — we 



might assume as before that two different functions 



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



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



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



(42) 



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

 equations were added together, the resulting equation would be 



/--xf(S4)--ixr[G"y 



+ 



^.y 



which is (35). In this case also, therefore, W is determined. 



'hdA =0, 

 (43) 



