18 



THE HUMAN MOTOR. 



So that, knowing the two forces and the angle which they form, 

 one can deduce the intensity of the resultant. 



When these forces are at right angles 



cos F If F 2 = cos 90 = ; 



and it follows that 



R 2 = F* + F* ; vide fig. 30. 



In the same way three or more 

 forces can be composed and their 

 resultant calculated. 



FIG 30 



H 



Conversely, a force, R, being given, it can be resolved into 

 two, or more forces. Two, on 

 the principle of the parallelo- 

 gram Three, on that of a par- 

 allelepiped ( 8) ; if there are 

 more than three directions, the 

 problem is indeterminate. Gen- 

 erally, the method of decomposi- 

 tion is that of three rectangular 

 axes. Let Fj, F 2 , F 3 , be the 

 three forces to be determined 

 along the axes X, Y, and Z, and ? 

 R the known force (fig. 31). *> si. 



By means of projecting planes, we form the parallelepiped of 

 ABCDEMHI, which will give the desired forces the intensities 

 F lf F 2 , F 3 . 



Notice that in the right angled triangle BMD : 



MD 2 = BM 2 + BD 2 ; 

 and in the right angled triangle BDC, we have : 



BD 2 = BC 2 + CD 2 . 



Therefore 



MD 2 = BM 2 + BC 2 + CD 2 , 



that is to say, that the square of the resultant is equal to the sum 

 of the squares of the components : thus : 

 R2 _ p 2 i -p* i ;p* 



J -I I * a I * 31 



or more generally 



R2 = X 2 + Y 2 + Z 2 . 



We also know that the directing cosines ( 3) can be written 

 F! = R cos a, F 2 = R cos p, and F 3 = R cos Y. 



If inversely the three forces were known it would be easy to 

 find their resultant, and by means of the directing cosines, to find 



