678 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This last equation is possible only if each member is constant, (— F), 

 whence ^ = Ti' ^^S (^V ■\- m) -\- a, ^ = — -p • log (PX -\- n) + h, and 



the u curves are a family of straight lines meeting in some point. No 

 other families of isothermal lines are possible curves of scalar functions, 

 the gradients of which are expressible in terms of the corresponding 

 orthogonal functions. 



The Jefferson Laboratory, Cambridge, Mass., 

 April, 1903. 



