On the Composition and Decomposition of Farces, 333 



kfeps them in equilibrio, passes through the point of con- 

 course, and is comprised in the plane determined by their 

 tlircctions. 



LEMMA. 



32. If a power P he applied to the circumference of a 

 circle moveable about its centre A (fig. 9), according to 

 the direction BP, which is a tangent to the circle at the 

 point B ; this power has the same tendency to turn the 

 circle about its centre, as if it were applied to any other 

 point C^ and in the direction CQ, which is also a tangent 

 at the point C. 



THEORE^f• 



33. When the directions of two forces P, Q, are in tht 

 same plane, and concur in the same point A (%. 10), if 

 we take upon these directions the right lines Af], AC^ pro- 

 portional to these forces, in such a manner that 



P:Q: : AB : AC; 

 and having completed the parallelogram AB DC; the 

 direction of the resultant of these two forces will be ac- 

 cording to the diagonal AD of the parallelogram. 



Demonstration. Conceive for a moment that the 

 point D is an immoveable obstacle, and from this print 

 upon the directions of the two forces let fall the perpendi- 

 culars DE, DF;. the triangles BED, CFD are similar, 

 because the angles at B and C being each of them equal to 

 the angle A, are equal to each other; therefore we have 



DC:DB::DF:DE; 

 & by the supposition P : Q : : AB : AC or : : DC : DB j^' 

 therefore, P : O : : DF : DE. 



From the point D as a centre, with the radius DF, de- 

 scribe the circular arc FG, meeting ED produced in G; 

 tlien, regarding this arc and the right line EG as inflexible 

 lines invariably connected to the point A ; conceive that 

 the force P is applied at E, in the direction EP, and that a 

 force M, equal to the force O, is applied to the point G, in 

 a direction parallel to AP, and consequently in the direc- 

 tion of a tangent to the arc FG ; and because the force 

 M=Q, and DF=DG, we have 



P : M : : DG : DE. 



Therefore (I8) the resultant of the two parallel force* 

 P, M, passes through the fixed point D, and is destroyed 

 by the resistance of this point; also these two forces are 

 in equilibrio about this point. 



Now the force Q^ the direction of which is a tangent to 

 the arc FG, and which we may regard as being applied to 



the 



