302 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Systems op Straight Lines and Circles in the xr Plane 

 WHICH satisfy Lame's Condition. 



If a set of curves {u = k) in the rx plane yield when revolved about 

 the X axis a set of isothermal * surfaces, the function 



^ (39) 



"u 



must be expressible as a function of u alone. The families of curves 

 which satisfy this condition are generally, of course, quite different from 

 those which satisfy the condition 



—r^ > a function of m, (40) 



'u 



for isothermal lines in the plane. A set of confocal conies with foci on 

 the X axis would, however, satisfy f both conditions. 



To determine what systems of straight lines in the xy plane satisfy 

 (39) we may represent any such system by the equation ax -\- fiy ■=. 1, 

 where a and y8 are functions of a single parameter u, and write 



a' 



9i 



9x a' 30 -{• (3' y 



9'u 



9x^ {aJx + ft'yf (a'x + ^'yY 



9''u _ 2/3/3' /3^(a"x + ^"y) 



9y'~ (a'x + /3'yy {a'x^/3'yf ' 



£(ll) _ 2 (gg^ + /3/3') a>'x + /3" y a(a^a: + /3' y) 

 V ~ a' + fi' a'x + /3'y y (a' + /3') ' ^ ^ 



In order that the sum of the last two terms of (41) may be expressible 

 in terms of u only, it is necessary that «' shall be zero, so that a is only 

 a constant, and the equation of any system of straight lines which satisfy 

 (39) is of the form 



* Lame, Leyons sur les coordonnees curvilignes, p. 32 ; Lemons sur les fonctiona 

 inverses, p. 5; Somoff-Ziwet, Tlieoretische Mechanik, I. 113 aud 138. 

 t Peirce, American Journal of Mathematics, 1896. 



