Composition and Decomposition of Forces. 29 



50. Let us suppose now, that while the directions of the two p . 

 forces p, q pass through the two fixed points K, N, their point of 14. 

 meeting A is removed further and further ; it is evident that the 

 sines of the angles BAE, DAE, BAD, will approach more and 

 more to a coincidence with the arcs CH, CG, HG ; if therefore 

 the point A is removed to an infinite distance from the fixed 

 points K, JV, CF, CI, HL, will coincide with the arc HG, 

 which in this case becomes a straight line perpendicular to the 

 two lines AK, AN, which are then parallel to each other and 

 to the line AE; and, since we have always 



p: q, 9 :: CI : CF: HL, 

 HL^CI+CF (fig. 13). 

 HL=CICF (fig.U). 



We conclude, therefore, that when two forces p, q, are exerted in Fig. is, 

 parallel directions, 



1. That their resultant is in a direction constituting another 

 parallel ; 



2. That if we draw a line FI perpendicular to these directions, 

 each of the forces will be represented by the part of this perpendicu* 

 lar comprehended between the directions of the two others ; 



3. That the resultant is equal to the sum of the two components, 

 when these act the same way, and to their difference, when their action 

 is opposed the one to the other* 



51. Since we have 



p : q : p : : El : EF : FI, 

 we have 



p : q : : El : EF, and p : 9 : : El : FI; 

 that is, of two parallel component forces and their resultant, ei- 

 ther two are to each other reciprocally, as the two perpendicu- 

 lars let fall upon their directions respectively from the same point 

 in the direction of the third. 



52. If we draw arbitrarily any line ABC, we shall have Geom. 



BC : AB : AC : : El : EF : FI, 

 and consequently 



p : q : p :: BC : AB : AC; 



that is, if a straight line be drawn at pleasure, cutting the directions 

 of two parallel forces and their rzsultant, each of these forces will be 



