676 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(1) If h = 0, 4>" (X) = 0, .//" (ix) = 0, cf> (X) =:cX +m, 



xp (fx) = d/x + n, and u =^ c X -{- djx + g. 



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

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



(2) If k ^ 0, ^' (A) = ^, ^' (,) = pJ^, 



^ (A) = ^, ■ log (P X + m) + a, x^, (/.) = ^ • log (^^ + «) + ^ 



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

 l^urely imaginary, the u curves are a set of parallel, that is, concentric, 

 circumferences. 



Every family of isothermal lines w^hich are tlie curves of a function u 

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

 concentric circumferences. No other families of parallel curves are 

 isothermal. 



3h 

 Thk Equation A„ • -^ = "^^(m)- 



du 



We have seen that equations (18) and (20) 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^, =f(v), and we may write 



q^ r—2pqs + pU = 0. (38) 



Monge's method yields the first integral ic — F I - ), (39) 



and of this equation 



„=,(:-±i) (40) 



is the complete integral and 



u^^i "t^ \ (41) 



where ij/ (a) — x — (a + p) ij/' (a), the general integral. 



Every family of straight lines in the xi/ plane, that is every set of 

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



