278 



M. Poinsot on the Percussion of Bodies. 



of the spontaneous axis corresponding to a centre of impulsion 

 whose coordinates are x and y. Other equally simple solutions 

 might be given, but it is not necessary to do so here. 



2. Considering t and u now as the coordinates of the sponta- 

 neous centre 0, which is in the line C G produced, we have the 

 relation 



t : u = x : y, 



which, combined with the preceding equation, gives 

 ci\v^ + ^y'" 



The coordinates of the spontaneous centre O are thus expressed 

 by means of those of the centre of impulsion C, or vice versa ; 

 for it will be observed that in these two expressions, as in the 

 symmetrical equations whence they are deduced, we may inter- 

 change / and X, and u and y ; in other words, the two centres C 

 and are always reciprocal. 



3. If H be the distance \'^x'^ + y^ of the centre of impulsion 

 C from the centre of gravity G, 

 and A the distance v/^^ + ij? of 

 the spontaneous centre from 

 the same point G, we shall 

 have, by putting for t and u 

 their values in x and y, and 

 forming the product AH of the 

 distances in question, 



(2? "1" Xj 

 AH=:« p 2 2 , ^2 2* 



Now, since the numerator and denominator here are each homo- 

 geneous functions of the second degree in x and y, we may 

 replace the latter by any other two lines proportional to them. 

 If, therefore, we agree that x and y shall now represent the co- 

 ordinates of the point D where the line C G cuts the ellipse 

 whose equation is 



and which we will call the central ellipse, the second term of our 

 preceding equation will be reduced to x^ + y^ simply ; but x^-\-y'^ 

 now represents the square 8^ of the semi-diameter GD deter- 

 mined in the central ellipse by the given direction of CG ; hence 

 between the distances A and H of the two reciprocal centres O 



