PROCEEDINGS OF THE AMERICAN ACADEMY. 



Two harmonic functions can only have the same level surfaces when 

 one is a linear function of the other. If upon n given portions, Si, S^, S.^, 

 . . . S,„ of a given closed surface, S, Fhas the constant values Oi, (7i, C3, 

 . . . (7„, respectively, ami V the values, F {C^), F (C^), F (Gs), . . . 

 F(C„), while upon the remainder of S, if there is any, the normal deriva- 

 tives of V and V are zero, and if V and V are harmonic within S, 

 V caunot in general be expressed as a linear function of V, and, if « is 

 greater than 2, their level surfaces will not usually coincide. If n is 3, 

 the condition of coincidence is evidently 



Ci F(00 1 

 C. F(C,) 1 

 C3 F(C,) 1 



= 0. 



(17) 



If U has the constant values Ci, Co, C3 ; V the constant values 

 Ki, K2, A'a ; and W the constant values Zj, Z.j, L^ on Si, S.,, S^^, re- 

 spectively, if the normal derivatives of these functions are equal to zero 

 at every point of S not included in Si, S.,, or S3, and if all these func- 

 tions are harmonic within S, W can always be expressed uniciuely in the 

 form A(/+ UV-\- Z>, unless 



Ci Ki 1 

 C\ K. 1 



c. a; 1 



= 0. 



(18) 



Before we were able to decide upon the forms and dimensions of our 

 apparatus and upon the manner in which it should be used, we found 

 it desirable to make some rather elaborate computations based on the 

 mathematical solutions of certain problems in heat conduction. In de- 

 scribing this work it will be convenient to state, first, some analytical 

 results to which we sliull afterwards give various piiysical interpretations. 

 We have purposely put these preliminary statements in purely mathe- 

 matical language lest they should seem to be narrower in their applica- 

 tions than they really are. 



(1) ,The square bases of a rectangular parallelopiped of height /are 2a 

 long and 2 a broad. A function, V, harmonic within this parallelopiped, 

 has the constant value V\ at the lower base and the constant value F, at 

 the upper base. At every point of the other faces of the prism V satisfies 

 the equation 



K^+/KF-F)=0, (19) 



