302 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Systems of 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 



L(u) 



-JTT (39) 



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 



V 2 (w) 



2 > a function of u y (40) 



"« 



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 -+- fty = 1, 

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



9x 2 (a'x + P'y) 2 (a'x + p'yY 



9^U 2 ft ft' P*(a"x + P"tr) 



9y 2 ~ (a'x + P'yy (a'x + {3'yf ' 



L(u) 2(aa / + ftft / ) a"x + /3"y a (a' x + (3> y) 

 ~hj r ~~ a 2 + ft 2 " a'x + fS'y y (a 2 + ft 2 ) 



(41) 



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, Lecons sur les coordonnces curvilignes, p. 32 ; Lecons sur les fonctions 

 inverses, p. 5; Somoff-Ziwet, Tlieoretisclie Mechanik, I. 113 and 138. 

 t Peirce, American Journal of Mathematics, 189G. 



