88 Statics. 



decomposed in the manner indicated by the figure, that is, each 

 into two parts, one in the direction of p, and the other perpen- 

 dicular to this direction. Then in the right-angled triangles 

 BAC, FAI, we shall have, radius being unity, 



Trig. 30. BC = AD = AB sin q AC, 



FI = AE = AF s'mrAC, 



AC = AB cos qAC, 



AI = AF cosr AC. 



Therefore, according to the principle above referred to, we shall 

 129. obtain, 4D -- ^E = 0; fr also AC+JII JG^O*, 



that i's, AB sin^ AC AF sin r AC = 0, 



and AB cos q AC -f AF cos r AC AG 0. 



The first of these equations gives 



AB sin q AC = AF sin r AC, 

 and consequently, 



AB : AF :: sin r AC : sin q AC, 

 that is, 



q : r :: sin r AC : sin q AC, 



143. which agrees with what was before demonstrated. A If the value 

 of AF be deduced from the first of the above equations, and sub- 

 stituted in the second, we shall have 



AB sin qAC cos r AC 



AB cos q AC H --- /- jrfT- -- AG = 0, 

 sin r J1C 



or 



jiBsinr.0Ccosg.tfC + AB sin qACcos rAC = AG sin r AC. 



But 



Trig. 11. sin r -AC cos 9 'ftC + sin 9 ^ C cos r ^ C = sin ( r A ^ + 



= sin 7*# r ; 



therefore, 



AE sin 9^ r = JIG sin r ^C ; 

 that is, 



: : sin r AC : sin q A r. 



