Dr. Stevelly on the Composition of Forces. 247 



their plane, and within the angle contained by their directions ; 

 and the resultant of any other two components which act respec- 

 tively in the same lines of direction and bear the same ratio to 

 each other, will be in the same direction, and will bear to the re- 

 sultant of the former pair the same ratio as that of either pair of 

 homologous forces of the two pairs of components. (1) For 

 let P and Q be two forces which act together on a material point 

 in directions making any given angle, a second P and a second 

 Q acting respectively in the same directions, would of course 

 have an equal resultant, say R, acting in the same direction 

 as before, and therefore 2P with 2Q would have a resultant 2R 

 still in the same direction. The same argument applies to 3P 

 with 3Q, 4P with 4Q, and nP with nQ, having for resultant nR 



P O P 



in the same direction still. Again, ^r with — must have -^ for 



resultant in the same direction ; for if not, the resultant R of P 

 with Q must be different either in direction or magnitude, or 



P Q 



both, from what it is. The same argument applies to -=. with ^, 



P Q 



and—, with —.. n' being any whole number, however large. From 

 n n 



this it readily follows (as in Euclid V.) that P' with Q' in the 

 same ratio as P to Q will have a resultant R' in the same direc- 

 tion and in the same ratio ; thus = = j^i = ^. But (2) if we 



vary the ratio by adding to either V or Q' the smallest additional 

 force, the direction at least of the resultant will be changed, 

 being brought from the previous direction nearer to that of the 

 increased force, as it must be the resultant of the previous re- 

 sultant and the added force. 



Corollary. If the two forces be equal, their resultant must in 

 this case manifestly have a direction which bisects the angle 

 contained by their directions; and any other pair of equal forces 

 acting in the same directions, but greater or less than those, will 

 have a resultant whose direction will still be that of the line bi- 

 secting the angle, and greater or less than the former in the 

 same ratio as the equal components 

 are increased or diminished. 



III. If two forces P and Q which act 

 together on a material particle are re- 

 presented in direction and magnitude 

 by AB and A C respectively, which con- 

 tain a right angle B A C, their result- 

 ant, whatever be its direction in the 

 angle B A C, must be such that its 

 square shall be equal to the sum of 



S2 



