M. Poinsot on the Percussion of Bodies. 353 



missible manner the couple to which it is due may have been 

 represented in the figure. In fact, the preceding expression (B) 

 shows this ; for by it, it is clear that the value of Q does not 

 depend upon the five particular values of P, a, b, a' and li', but 

 solely upon the values of the two products P(« — ft') and P(6 — 6'), 

 which are the respective moments of the proposed couple relative 

 to the axes of y and of x. On this account we shall find it more 

 convenient to preserve these moments, solely, in the formula (B), 

 and to represent them by the simple letters L and M, which 

 latter will suffice to define at once the magnitude and the posi- 

 tion of the given couple under consideration. The expression 

 (B) is thus replaced by the more concise one 



^~u^x^ + l3Y + »^l3^' • • • • ^"^ 

 in which nothing, beyond the necessary data of the question, is 

 visible. 



Before proceeding further, a brief remark still remains to be 

 made. 



80. In art. 28 we have seen how, from the formula (A.) rela- 

 tive to the impulsion of a single force, the formula (B) relative 

 to the impulsion of a couple may be deduced. Now it may be 

 shown that, conversely, from the latter, supposed to be known 

 — and it would be easy to demonstrate it directly — we may also 

 deduce the former. In fact, let it be required to find the per- 

 cussion Q which the body is capable of producing at any point 

 D, whose coordinates are x and y, in virtue of a single impulse 

 P which the body has received at the point C, whose coordinates 

 are a and b. We may consider the simple force applied at C to 

 be decomposed into an equal, parallel and like-directed force 

 applied at D, and into a couple (P, — P) applied to the arm CD. 

 Now the force applied at the point D will evidently there cause 

 a percussion equal to P ; and according to the formula (B), the 

 couple, on its part, will cause a percussion at the same point D 



equal to 



^ »\a-x)x + l^{b-y)y 



The required percussion Q, therefore, being the sum of these 

 two, is 



this is precisely the formula (A) which was demonstrated at the 

 commencement. 



Let UK now return to the case where the body is animated by 

 a couple only. 



