KUKIKS ON A I-AUTP 'I.!.. 



28. The expression for the magnitude of the resdtaat ta 

 .ny be rendered independent of the poatioft of tb 



a tho expressions on the right-hand side MB 

 we shall find that tho coefficient of P is 



cos'a, 

 and that tho coefficient of PP is 



, t 



we know from Analytical Geometry of three 



and that 



co a, cos , -r cos ff t co 0, + cot 7, co< 



is equal to the cosine of the angle between the directions of 

 :.>rces P, and P t , which we maj denote by ow(P 

 !ar values will M 10 coefficients of the other 



:s; and the result may be expressed thus, 



i?-2P i + aSPP'cos (P, P*) 

 where by P, P' we mean any two of the forces, 



29. The equation 7?cosa-2Peo*, in Art W, shews 



the retolrttl part of the retw/tonl in any Jt'rrrtfo* u <y*al 



to the turn of the reiolved parti of ll comfmvti* m th< *am< 



'ion; for since the axes were taken arbitrarily, that of x 



been made to coincide with any assigned dirte- 



Or we may establish the projiosition than. Sanpose 



a straight line drawn through r of application of the 



d to the axes at angles a', /?, 7*. Take the 



equations of Art 26, 



R cos a - P cos a, 4- P. cos a, 

 R cos b - P, cos/9, + P.cos/9, 



COS C - P, COS 7, + P.COS7, 



