F R A 



with A y angles ft, /3, (3 3 



* 3 



with A wangles 7, ^, y y 



i s w 



and make 



p COS. -}- jtf COS. +/> COS. a, -f- p COS. a =: X 



112 233 n n 



p cos. ,3 -f p cos. /3 -f ;> cos. /3 + j cos. = Y. 



1 1 2 2 3 3 ?J 7i 



p COS. y -f-P COS. y -i- p COS. 7 -\- p COS. ^ Z 



112833 n n 



the forces will be equivalent to X in A .r, Y in A j/, and Z in A^. 



If R be the resultant, and 9, v, the /s. which it makes with A r, Ay, 

 A # respectively, we shall have 



R = v ' (X* + Y* + Z) 



. X Y Z 



cos. & , cos. j = -^, cos. ? -^ 



One of the three last Equations is superfluous. 



7. When a point is acted upon by any forces, to find the conditions of 

 equilibrium. 



In order that there may be an equilibrium, the resultant of all the 

 forces must be o. And in order that this may be the case, it is evident 

 we must have in Art. 5, X = o, Y = o; and in Art. 6, X = o, Y = o t 

 Z o. Hence we have for the conditions of equilibrium in the former 

 *ase 



P COS. K, +p COS. a, -I. p COS. a. 4- ~ O 



1 12 2 ^3 3 



p sin a. 4- P sin. et. -4- p sin. ex, -4- =: o 



1 12 23 3 



And in the latter case 



p cos. u. -\-p cos. u, -4- p cos. a, 4. = o 



1 1 1 2 83 3 



p cos. /3 + p cos. /9 4- p cos. 8 + ...... = o 



1 12 23 3 



p cos. y -{- /> cos. y 4- P cos - y + 



FORCE. Sec iVo^iw//. 



FORCE mating, or motive. .SVt 3 Momentum. 



FORCES, centripetal and txntrifirgalrSee C&ifrat'Itorvet. 



FR ACTIONS continued 



Continued fractiww ar- very useful when we have a fraction or ratio 



