674 PROCEEDINGS OP THE AMERICAN ACADEMY. 



If — j-^- is a function of u only, the last term in the second member 



of (30) must be expressible in terms of u only. If a' — ft' = 0, we 

 have a family of concentric circumferences. In general we may write 

 a" : a' = ft" : ft' = y" : y', or ft = m a + n, y =z 2 k a + I, so that the 

 equation of the circles must be of the form 



x"" + y- — 2 ax — 2 1/ (m a + n) — 2 k a - I = 0, (31) 



where a is the only parameter. If we represent the first member of this 

 equation by Sa , the equation Sa^ — Sa^ = represents the straight line 

 through the points of intersection of the circles which correspond to the 

 two values ai , ag of the parameter. In this case the line is a; + 7??y + ^ = 0, 

 whatever the values of ai and oj, therefore, as is well known, the system 

 of isothermal circles * must pass through two fixed, real or imaginary, 

 points. 



Functions the Gradients of which are expressible 

 IN Terms of the Functions themselves. 



Several of the conditions stated in the previous pages [see (d), (h), 

 and equation (18)] require that the gradient of a function be expressible 

 in terms of the function itself, so that the normal derivative of the 

 function has the same numerical value at all points of any one of its 

 curves of level. We may state this requirement in a somewhat simpler 

 form, however, if we remember that since the gradient of any function, 

 </), of u is equal to (f>' (u) • h^, the lines of all functions which satisfy the 

 equation /?„ =f(u), whatever f may be, are included in the lines of 

 functions which satisfy the equation h„ = k, where k is any constant (for 

 instance 1). Every such family of lines forms a set of parallel curves. 

 "We have to solve, then, the equation 



one of the standard forms for partial differential equations of the first 

 order. 



Its complete integral is 



u =^ ax -\- y \/k'^ — a- + c, 

 and its general integral, 



* Darboux, Lemons sur la tlieorie gencrale des surfaces. 



