v MATTER IN RELATION TO MOTION 65 



board. Obtain two pieces of thin india-rubber cord twenty 

 inches long, and fasten small loops of string to the two ends. 

 Pin one of these loops to the board so that the upper end of 

 the india-rubber cord coincides with the zero of the scale. 

 Attach the upper end of the other india-rubber cord to any 

 convenient point on the board. Bring the two lower ends 

 together and hook on to them a weight (hay of 100 grams). 

 Measure off. twenty inches from the upper end of each cord. 

 The excess of length in each cord will be proportional to the 

 tension of that cord. Complete the parallelogram with chalk, 

 and show that the diagonal is vertical and is equal to the 

 extension of the cord when it hangs vertically by the side of 

 the scale with the weight attached. 



These experiments lead to the rule which is always known as 

 the parallelogram of forces. It is usually stated thus : If two 

 forces acting at a point be represented in magnitude and 

 direction by the adjacent sides of a parallelogram, the re- 

 sultant of these two forces will be represented in magnitude 

 and direction by that diagonal of the parallelogram which 

 passes through this point. 



A line may be drawn to graphically represent a force, its 

 length being made proportional to the force, and its direction 

 showing the direction of the force. 

 Forces can thus be represented, both 

 in magnitude and direction, by lines. 

 Let O represent a material body acted 

 upon by two forces, represented both 

 in amount and direction by the lines 



OB, OA. To find the resultant of Plo . 25 ._ The p arallelogram 

 these two forces as the single force of Forces. 



which can replace them is called, both 



as regards its amount and direction, we complete the parallelo- 

 gram OBRA and join OR, which will be the resultant required. 



Calculation of Resultant. When the two forces whose re- 

 sultant is required act at right angles to one another, the calcu- 

 lation is a simple application of a proposition in the first book of 

 Euclid (I. 47). Under these circumstances the triangle OBA 

 is right-angled, and Euclid proves that (OA) 2 + (AR) 2 = (OR) 2 , and 

 consequently (OA) 2 + (OB) 2 = (OR) 2 , from which when OA and 

 OB are known we can calculate OR. 



When the directions of the two forces OB and OA are inclined 



