If ill these expressions we replace h by its value — , and P by its 



a 



174 M. Poiusot on the Percussion of Bodies. 



If ill these expressions we 

 value Mff^, we shall have 



(« + . 

 K2 





^ = («-t)^-^ 



a^2 + K2^ 



now a+ar being the distance of the point C from the sponta- 

 neous centre of rotation of the body, {a-^x)6 is the actual velo- 

 city of this point C ; hence, since the percussion Q at this point 

 is measured by the product of its velocity into a mass 



we may say that the point C strikes with the same force as it 

 would do were this fractional part of the whole mass M there 

 concentrated. 



Similarly, the factor (a -\9 in the second expression being 



the velocity of the point 0', we see that this point, reciprocal to 



the first, strikes as if it were charged with the mass n = M-^5 — =yg, 



in other words, with the same force as that with which it would 

 strike if this other fractional part of the mass M were there con- 

 centrated. 



But it is at once manifest from the above values of the masses 

 m and n, first, that their sum m + n constitutes the whole mass 

 M of the body, and secondly, that they are to each other in the 

 ratio 



that is to say, they are inversely proportional to the distances of 

 the two points C and 0' from the centre of gravity G of the 

 body. 



Considering, therefore, any two centres, reciprocal to one 

 another, and the respective percussions they are capable of pro- 

 ducing, we may say that two such points strike exactly as if the 

 whole mass of the body were divided between them in shares 

 inversely proportional to their distances from its centre of gravity. 



27. We may, consequently, conceive the body to be replaced 

 by the right line CO', considered as an inflexible rod without 

 mass, but loaded at its extremities with the two material points 

 m and n in question. This rod would be gifted, not only at its 

 two extremities, but also at every other point of its direction, 



