28 Statics. 



48. We remark further, that since the two component forces 

 5,6. jo, q, being represented by the two sides AD, AB, of the paral- 

 lelogram DABE, their resultant must necessarily .be represent- 

 ed by the diagonal AE of the same parallelogram ; by calling 

 Q the resultant, we shall have 



p : 9 :: AD : AE, 

 q : p :: AB : AE ; 

 that is, 



p : q : p : : AD : AB : AE, 

 :: BE : AB : AE. 

 Now in the triangle ABE, we have 



BE : 4B : AE : : sin BAE : sin BEA : sin ABE. 

 But on account of the parallels BE, AD, the angle BEA = DAE, 

 64. and the angles ABE, BAD, being supplements to each other, 

 Tri S- 13 - sin ABE = sin BAD ; hence 



BE i AB i AE i: sin BAE : sin DAE : sin BAD ; 

 and consequently 



p : q : $ : : sin BAE : sin DAE : sin BAD ; 



from which it will be seen, that if we suppose the force p ex- 

 pressed by sin BAE, the force q will be denoted by sin DAE, and 

 the force g by sin BAD; that is, the two component forces and 

 the resultant may be represented each by the sine of the angle com- 

 prehended between the directions of the two others. 



In representing forces, therefore, we may employ indifferent- 

 ly either the lines taken in the directions of these forces, or the 

 sines of the angles comprehended between these directions, pro- 

 vided we take for each the sine of the angle comprehended be- 

 tween the directions of the two others. 



This last method of expressing forces has its peculiar advan- 

 tages, as we shall see in what follows. 



Fig. 11, 49. If from the point A as a centre, and with any radius AiC, 

 we describe an arc of a circle HCG, meeting in G and H the 

 directions of the forces p, q, and let fall from the point C upon 

 J1D, A.B, the perpendiculars CF, CI, and from the point H 

 upon J1D the perpendicular HL, it will be readily seen that CF, 

 CI, HL, are the sines of the angles DAE, BAE, BAD, respec- 

 tively ; we have accordingly 



p : 9 : 9 : : CI : CF : HL. 



