Compound Motion. 23 



AF, contribute nothing to the resultant, because they act in op- 

 posite directions, and are equal to each other. Indeed it will 



be readily seen, that, from the nature of a parallelogram, the 32, 

 two triangles DGA, EIB, are equal ; therefore DG = BI, and 

 consequently AF AH. 



As to the two forces AI, AG (fig. 5.), since they are exerted ac- 

 cordins: to the same line, and are directed the same way, the result 

 must be the sum of the two effects AG, AI ; and in fig. 6, since AI, 



AG, are exerted according to the same line, and are opposed the 

 one to the other, the result must be the difference of the two effects 

 AG, AI. But as the triangle EIB is equal to DGA, we shall 



have (fis. 5.) 



JlI-\ AG=;AI + EI=AE; 

 and ( fig. 6.) 



AI AG=AI EI = AE. 



We conclude, therefore, that the four forces AF, API, AG, AI, and 

 consequently the two forces AD, AB, have no other effect, than 

 the force AE, represented by the diagonal of the parallelogram 

 DABE, of which the two sides AB, AD, denote the forces q, p. 

 This proposition is known by the name of i\\e parallelogram offerees. 



39. We have in what precedes, represented the two for- 

 ces p, q, by the lines AD, AB, which they are capable Fig. 4,5, 

 of making the body m describe in the same time, that is, 6 - 

 by the velocities which they would communicate ; although, 

 according to what we have said, the true measure of any 28. 

 force is the quantity of motion that it is capable of producing. 

 But as the quantities of motion are in the ratio of the velocities, 

 when the mass is the same, as is the fact in the present case ; 29 

 we may always, as we have now done, take the velocities AD, 

 AB, as representing the two forces. 



But if, instead of having immediately the velocities which the 

 two forces p, q, are capable of giving to the body m, we had the 

 quantities of motion which they would produce in known masses, 

 we should take AD, AB, in the ratio of these quantities of motion. 

 If, for example, I know the forces p, q, only by this circumstance, 

 that the force p is capable of giving a known velocity u, to' a 

 known mass n ; and that the force q is capable of giving a veloc- 

 ity D to a known mass o ; I should take 



AD '/AB ::'nw : er. 



