7 
when the sides of the 
_ given forces and their 
in the same direction 
THE POLYGON OF FORCES 35 
Sieh two forces and their equilibrant are represented, the three arrow- 
heads will point in the same direction round the triangle. 
Referring to Fig. 28, OR?=OP?+0Q?+20P+OQ cos POQ, also 
: AC AB + BC? - 2AB+ BC cos ABC. 
54. Polygon of Forces.—If a number of forces OP, OQ, OR, and 
4 os act at a point O (Fig. 29), their resultant or their equilibrant may 
be found by repeated 
application of the 
parallelogram or tri- 
angle of forces. The 
weet POQT 
nes OT, the 
resultant of OP and 
OQ. The _parallelo- 
gram TORU deter- 
mines OU, the re- 
sultant of OT and 
OR, and _ therefore 
the resultant of OP, Fig. 29. 
OQ, and OR. The 
parallelogram UOSV determines OV, the resultant of OU and OS, and 
therefore the resultant of OP, OQ, OR, and OS. 
If AB be equal and parallel to OP, and BC be equal and parallel to 
OQ, then AC will be equal and parallel to OT; if CD be equal and 
parallel to OR, then AD will be equal and parallel to OU ; and if DE be 
equal and parallel to OS, then AE will be equal and parallel to OV. 
Hence if a polygon ABODE be drawn, having its sides respectively 
allel and equal to the given forces, the closing side AE will represent 
in magnitude and direction the resultant of the given forces, and EA 
will represent their equilibrant. 
If arrow-heads be placed on the sides of the polygon of forces to show 
_ the sense of the forces, then when the sides of the polygon represent the 
given forces and their resultant, the arrow-head for the resultant will 
point in the opposite direction round the polygon to that of the other 
arrow-heads. But 
polygon represent the 
equilibrant, all the 
arrow-heads will point 
round the polygon. 
When the given 
forces do not all act 
at the same point, or 
when their lines of 
action are not concur- 
rentand notall parallel, 
their resultant may still . Fie. 30. 
be determined by re- 
peated application of the parallelogram of forces, as shown in Fig. 30. 
