156 PROCEEDINGS OF THE AMERICAN ACADEMY. 



equation (31) is satisfied and 



da „ drjT 1 „ , V 



eJ = «' a7 = ? ^ = '^- ^'*^ 



In this case, if we put c= 1, ^ and /j. are the Stokes functions corre- 

 sponding to a and A. If the level surfaces of the harmonic space 

 functions, V and W, are surfaces of revolution about two different 

 straight lines in the .ri/ plane, the functions a and A which represent 

 the values of Fand W in this plane do not in general satisfy (31). 



Graphical superposition of the lines of force in the a-i/ plane due to 

 an infinitely long, homogeneous cylinder of revolution parallel to the 

 axis, and to a homogeneous sphere with centre in the plane, will not in 

 genera] yield the lines of force in the a:i/ plane due to a combination of 

 the two masses. 



If a and X are harmonic, any linear function (but no other than 

 a linear function) of a is harmonic, and any two linear functions of 

 a and A satisfy (31). There generally exist, however, non-linear 

 functions of a and A which, although they are not harmonic, satisfy the 

 condition. The functions {w^ — y"^), {x^ + 3/^)", the second of which is 

 not harmonic, obey (31), as do the harmonic pair {x"^ — y"^), log (x^ + ?/^). 



As a simple example of the fact that a harmonic function and a 

 function which is not even isothermal may satisfy the condition (31), 

 we may consider (2 y"^ — x'^) and (y^ — x"^). 



The non-isothermal functions x"^ — ay"^, y"^ — ax"^, which are solu- 

 tions of the equation 



d^_^d^_^V__^V_^^ (35) 



dx^ dy"^ X ' dx y ' dy ' 



evidently satisfy the equation (31). 



If a and A are any two solutions of the equation 



where f{x) is any given function of x, the condition (31) is satisfied 

 and zj z=f(x). 



