84 A. Mukhopadhydy — Bemarlcs on Mongers Equation. [Feb. 



Also, 



UV2 



Ci C2 — 



- (2) ■ 



Hence, finally, we have 



_ 1 T2 



""■ U* 





This, therefore, is the differential equation to which in reality belongs 

 Col. Cunningham's interpretation. I may mention in passing that when 

 e^ = 2, we have T = 0, and, when e^ = 1, we have U = ; so that, T = is 

 the differential equation of all equilateral hyperbolas, U = of all 

 parabolas, V = of all pairs of right lines, and, W = of all central 

 conies. I may also remark that I have never seen the eccentricity thus 

 expressed in terms of the differential coefficients. Also, since 



2 

 7r3 



(Area) ^ = 



U 



and, Ci= 



we have, 



27 

 Area = — 



'Off 



and, I have never seen the area expressed in this form.* 



* Of course, the two absolute invariants, viz., the area and the eccentricity, may 

 be expressed in terms of the radius of curvature p, and arc, s ; thus, we have 



27 TT p2 



Area = 



("©■ 







.M-,-.,ff 



& 



