A CERTAIN QUADRATIC FORM 241 



If pi is the perpendicular frona M on the direction of Fi, the moment of Fi 

 about N is, A being the area of ABC, 



'"--(?-?)k- ■ <^^' 



ri^cj = X-co(ri2),-, 



= (ri-w)i = constant, (26) 



with like values for the moments of Fo, F3. Since any position of the 

 figure can be brought to any other position by a pure rotation followed 

 by a similitudinous projection from M, each line of the figure turns through 

 the same angle in the same time. Let co be the angular velocity at any 

 instant then, 



STOiri^o) = constant 

 X^o) (Smjri^); = a)i(Smiri^)t. 



Hence 



Therefore 



X-oj = coi = constant. (27) 



= (ri^i^)i = constant, (28) 



with like values for r2^cxi, r^ co. Differentiation -with respect to t gives 

 Pi^i = ViPi = PiFs = 0. 



Hence must 



X y z 

 mi m-i ms 



(29) 



or the force-center must coincide with M the mass-center, and 



hi fn /is 



niimsa nizmib m^miC 



In particular if the mutual forces are proportional to 



then must a = h = c, which is Lagrange's theorem of "the three bodies" 

 when n = —2. 



In this movement 



ds\ dsi ds 



