234 rNIVERSITY OF VIRGINIA PUBLICATIONS 



Replacing- the k's by m's (6) becomes 



The moment of inertia of a system of masses about a point is equal to 

 the moment of the mass M of the system at the mass-center plus the mass 

 mean of the products of the moments two at a time of the masses about 

 each other. When p = the last term on the right of (8) is equal to the 

 moment of inertia of the system about its center of inertia. We may speak 

 of it as representing the internal moment, of inertia of the system. 



4. Let Q =. X dy — y dx = p-dB. 



If the points Pi move in space, being arbitrarily fixed and the k's 

 remain constant then the point P is homogeneously displaced with the 

 system P,-. Let the projections of these points on the arbitrary a;2/-plane 

 describe paths whose elements of sectorial area are represented by d{P), 

 d{Pi). Represent by d(P,-,) the corresponding area swept over by a 

 radius vector parallel and equal to the projection of P,P,- on the a;2/-plane. 

 Then 



d{P) = Zki d{Pi) - 'S.kiki d{Pi,). ■ (9) 



If the paths are closed so that the points return to their initial positions 

 then the areas bounded by them are related by* 



(P) = 2fc,- (P.) -^kiki {Pi,). (10) 



This is the generalization of Elliott's extension of Holditch's theorem. 

 It is obvious that the areas for any three points and those of their relative 

 motions are all that are necessary for a motion of homogeneous displace- 

 ment in a plane in order to determine the area for any other point. Thus 

 if Pi, P2, P3, the corners of a triangle move in a plane, then any point P, 

 moving in the plane homogeneously with them, will have its path area 

 given by 



P = x{P,) + y{P,) + z{Pi) - xy{P,,) - yz{P,,) - zx{P,,), (11) 



where x, y, z are the areal coordinates of P. If P be constant the locus of 

 points which describe equal areas lie on a conic, and different values of 

 (P) give a system of similar and similarly situated conies. The areas in 



* If the points Pi move completely around the boundary of the same simple 

 closed curve and their mutual distances are so related that Sfe, hj (Pa) = then will 

 the point P trace the same curve. 



