250 Dr. Stevelly on the Composition of Forces. 



A B' with B' D 2 ; and then the two resultants could not be one 

 and the same force, as we have proved them to be; and no 

 other line but AD 2 can represent the direction of both, nor 

 therefore of P with Q when represented each by A B and B D 2 

 respectively. 



(3) It may be remarked that the line AD 2 bisects the angle 

 Dj A C. Then, if again we draw A D 3 bisecting the angle D 2 AC, 

 it can be shown by a precisely similar argument to that already 

 employed in (2), that AD 3 must represent, both in direction 

 and magnitude, the resultant of two forces P and Q represented 

 by A B and B D 3 . This is done by producing A D 2 and letting 

 fall the perpendicular D 3 B" on A D 2 produced, and then using 

 what we have now proved in (2), as in proving (2) we used what 

 had been before shown to be true in (1), and so on without limit, 

 to any number of bisections. If we again bisect D 3 A C by a line 

 A D 4 , it can be shown to represent the resultant of P and an- 

 other Q, represented respectively by A B and B D 4 , and so on to 

 A T> n , n being any number, however large. 



(4) Let now P and Q be any two forces whatever acting- 



together in directions which contain a right angle on the mate- 

 rial point at A, represented respectively by A B and Bz. Then 

 shall A z represent both in magnitude and direction the result- 

 ant of the forces represented by A B and B z ; for z, whatever 

 be the magnitude of Q, must fall between some two consecutive 

 intersections of the bisectors of the successive angles used in (2) 

 and (3) with the line B z, say between D w _ x and ~D n (n being 

 any whole number of the series 1, 2, 3, ... n). Draw Ax bi- 

 secting the angle D»-i A D n , and from x let fall the perpendi- 

 cular xB' onAD^-1 produced, and xB" on AD W . Now the 

 triangle D W _!#B' being equiangular with ABD B _,, and it 

 having been already proved that AD W _ X is the resultant of AB 

 with B J) n -\j D w _! x is also the resultant of D w _j B' with B' x. 

 If, then, along with AB and BD^_i we introduce D n _ 1( 27, and 

 with A D ra _ l5 D n _i B' with B'x, the first three, or their equiva- 

 lent A B with B x } are equivalent to the last three, or their equi- 

 valent A B' with B' x. In a similar way, since the triangle 

 x B" D n is equiangular with A B D n) and it has been already 



