i62.] STABILITY. 97 



co-ordinates are found by solving these equation for f and 77, or 

 .e the coefficient of cos <f> in (2) vanishes by (i), by solving 

 -.he equations 



(3) 

 '(4) 



Putting, for shortness, V(2X) 2 + (Y) 2 =R, t(xY yX) H, 

 2,(xX+yY) = K, we find the co-ordinates of the centre, 



"= 



161. By the rotation of the figure, the magnitude of the 

 resultant R of the system is of course not changed. But 

 the resulting couple H for the origin, or what amounts to the 

 .same, the moment of the system about the origin, is changed and 

 becomes, by Art. 160, 



H 1 = 2 [(x cos (/> ~y sin <f>) Y (x sin < +y cos 

 yX) - cos </>-2 (xX+y F) - sin 

 K sm </>. (6) 



This couple //"' vanishes if the figure be turned through an 

 angle </> determined by the equation 



tan<=^. (7) 



162. If the system of forces be originally in equilibrium, we 

 have 2X=o, 2F=o, ^(xY-yX}=o (Art. 143). Hence after 

 turning the figure through an angle <, the forces will be equiva- 

 lent to the couple 



H'=-Ksin<l>. (8) 



This couple has its greatest value when = ?r/2; it vanishes 

 only when < = TT, in which case the system will again be in 

 equilibrium. 



PART II 7 



