TO SEVERAL CENTRES. 



195 



P bisects a a' and is PC, and produced, P M cutting the 

 axis. From hence may be seen how complicated would be 

 the analysis, how next to impossible the geometrical con- 

 struction of the locus of P, by referring the lines P M to 



771 



S S' as an axis. We know indeed that one of the forces 3- 



r 2 



771 



or - acting towards S or S', the locus of P is an ellipse; 



but it would not follow that if both forces acted the same 

 curve would be the locus. That the force would be differ- 

 ent is certain, because it would be as P c, and not as either 

 P a or P a'. But it may be said that the curve also would 

 be different. Let us, however, suppose the case of the 

 curve, whatever it be, cutting the axis S S' produced at 

 I and I', points equally distant from S and S', so that 

 S I = S' I' ; also that the angle and the initial velocity of 

 projection from I and I' is the same, and further that the 

 attraction as the mass is the same from S and S', or that 

 the mass of the body in S and in S' is the same ; then it 



o 2 



