676 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(1) If k = 0, 0" (X) = 0, f 0) = 0, (A) = c\ +m, 



\p 0-0 = <£/*• + n, and «< = c X + efy* + g. 



If c and ± d are either real and equal, or conjugate complex quantities, 

 the u curves are a set of real parallel straight lines. 



(2) If* t 0, il{S>=w l— i *'(,)=^ ( 



■A00 = ^ ' log (* a * + »0 + «. *0) = p • log (**/. + ») + i, 



and if the constants of integration are so chosen as to make u real or 

 purely imaginary, the u curves are a set of parallel, that is, concentric, 

 circumferences. 



Every family of isothermal lines which are the curves of a function u 

 which satisfies (18) is either a set of parallel straight lines or a set of 

 concentric circumfereuces. No other families of parallel curves are 

 isothermal. 



9h 

 The Equation /*„ • V-' = V 2 (u). 



du 



"We have seen that equations (18) and (2G) are equivalent; this equa- 

 tion, therefore, defines the families of straight lines which form the 

 orthogonal trajectories of the families of parallel curves defined by the 

 equation h v =f(v), and we may write 



q* r—2pqs + p 2 I = 0. (38) 



Monge's method yields the first integral u = F ( - ), (39) 



and of this equation 



is the complete integral and 



a + y \ 



{a) - xj' 



- * C^V-- )■ 



where \f/ (a) — x = (a + y) if/' (a), the general integral. 



Every family of straight Hues in the xy plane, that is every set of 

 lines defined by the equation ax -\- py = 1, where a and p are arbitrary 



