FOR 



Cor. 5. If the ^ between two given forces be diminished, the resultant 

 is increased. 



Cor. 6. If any number of forces be represented by sides of a polygon 

 taken in order, their resultant will be represented by the line which com- 

 pletes the polygon. 



Cor. 7. A number of forces which are represented by all the sides of a 

 polygon taken in order, acting upon a point, will keep it at rest. 



3. If the edges of a parallelepiped drawn from the same point, repre- 

 sent three component forces, the diagonal will represent the resultant. 



Cor. 1. If any number of forces be represented by sides, taken in order, 

 of a polygon, which is not in the same plane, their resultant will be re- 

 presented by the line which completes the polygon. 



Cor. 2. If any number of forces be represented by all the sides, taken 

 in order, of a polygon, they will keep a point at rest. 



4. To find, by means of equations among the symbols, which the forces 

 and their positions introduce, the resultant of two forces acting at a 

 point. 



If we suppose a line, as A x, to pass 

 through A, we may determine the posi- 

 tions, both of the components and resul- 

 tant, by the /s. which they make with 

 this line. 



Let p and q be the forces in A P, AQ -, 

 a,, (B the /s. which they make with A x. 

 Resolve p into two forces in the direc- 

 tions A x t and A y perpendicular to 

 A x t then the resolved parts will be p -A. 3VI X. 



cos. , p sin. <*. In like manner q is equivalent to q cos. /3 in the direc- 

 tion A or, and q sin. /3 in the direction Ay. Hence the forces are equi- 

 valent to 



p cos. , q cos. /3 in A x. 



p sin. , q sin. (Sin Ay. 



And the resultant of p and q will be the resultant of these four forces. 

 If we put 



p cos. a, + q cos. /3 = X. 



p sin. -f- q sin. /3 = Y. 



and take in A x, Ay, A M = X, A N = Y, and complete the rectangle 

 A M R N, A R will be the resultant ofp and q t and if r be this resultant, 

 and 8 the / which it makes with A x> we have 

 122 



a 



