236 xmiVEESiTY of Virginia publications 



Kempe's theorem, that points describing equal areas lie on a circle (14), 

 and for different values of (P) we have a family of concentric circles is now 

 obvious. Using r, ri, r^, rz as the mean radii of the areas (P), {A), (B), 

 {€), then the circle. 



= l,x n^ - 'Exy c^ (15) 



is the orthogonal circle of the three circles with centers A, B, C, and radii 

 n, ra, rs. The area enclosed by the path of any point x, y, z as given by 

 (14) has for the square of its mean radius r- the power of x, y, z with respect 

 to the circle (15). The minimum area enclosed by the paths of all points 

 in the plane is that for the center of (15) and is equal to the area of that 

 circle. 



When in the interpretation of these problems a circle becomes a straight 

 line by reason of its radius becoming infinite, the power of a point with 

 respect to the circle becomes the power of the point with respect to the 

 straight line or the perpendicular distance of the point from the line. The 

 proof of this is general for the circular locus in any space, we give it for the 

 plane. 



The equation of a circular locus is 



p = 'Zxpi — S xyc^. 



The power of any point ■with respect to a circle is equal to the product 

 of its distance from the center d and its distance d from the polar with 

 respect to the circle, hence 



when the radius becomes infinite the last term is zero and the limit of 

 di/d, etc., is unity. The circle passes over into the line and 



5 = 'Sxdi. 



Taking up (10) again and replacing k's by m's and dividing by dt, 

 there results 



'-^ dt dt M -^ ' dt ^ ^ 



The moment of momentum of a mass system is equal to that of the sys- 

 tem concentrated at its mass center plus the internal moment of momentum 

 of the system. If the mass center is at rest the external and internal 

 moments of momenta are equal. The equation (16) was first derived by 

 Laplace {Mechanique Celeste, Liv.I, Chap. V, § 22), in his investigation of 



