TWO EQUAL MASSES OF OPPOSITE SIGNS. 137 



If / be the inclination of the force F with the tangent, and A 

 the complement of the angle o>, we have 



(21) tan /= -^ = 2 cot w= 2 tan A. 

 We get lastly, for the force itself, 



F 2 = X 2 + Y 2 = F 2 B + F*=/^ -Y( 3 cos 2 <o+i) 



(22) ^''-sin'A+i) 



154. Prolong the tangent as far as the axis at T, and the direction 

 of the force to S ; the triangles OPS and OPT give 



PS OS 



sn <o cos 

 PS ST 



from which we have 



cos i sin / 



and finally 



OS tan /= ST cot o> = ST tan A. 



As tan i=2 tan A, we see that 



ST = 2OS, 

 whence 



(23) OT = 3 OS. 



This theorem is due to Gauss. 



The value of the force is easily expressed as a function of the same 

 lines. We have, in fact, 



R = OT cos w = 3 OS cos (o, 

 = Rcosw, 



and, consequently, 



,2,.,. 



cosw = 



OS 



