102 



Dr. G. H. Bryan and Mr. W. E. Williams. [Jan. 7. 



- g sin d Q 80 - X u 8u - X v 8v - XeS0, 



- g cos 6 Q 8d - Y u &u - Y v 8v - Y*S<9, 



d « , d 



To solve these equations, put 



8u = Pe w , St> = Qe*< 



8(9 = Be 



Substituting, dividing by e xt , and re-arranging the terms, we have 

 -p(X + X u ) + qX v + -R(XXe - \v +gsm6 ) = 0, 

 PY tt + Q(X + Y t ,) + E(XYj + Xu + g cos ) = 0, 

 m u + QG v + R(\W + \Ge) = 0. 



Eliminating P, Q, E, 



X + X u , X„ , - Ai' + AX<j + # sin # 

 Y u , X + Y v , Aw + XYe + g cos 6 

 G u , G y , X 2 Je 2 + XGe 



= (4). 



If the determinant be expanded in powers of X we get an equation 

 of the form 



AA4 + BA3 + CU 2 + DA + E = (4a),. 



where 



E = 



Jc 2 (X u + Y v ) + G e , 



¥ (X V J V - X V Y U ) + v Q G u - u G v - Xe G u - Ye G,, 

 uq (X P G M - X U G V ) + vq (Y v G u - Y U G^), 



- g sin 6 G U - g cos G V + Y<j (X v G ?i - X it G y ) 



- Xe (Y V G U - Y.G.) + G (X U Y V - X V Y V ), 



g cos (X v G tl - X U G V ) - g sin (Y V G U - Y tl G v ), 



In many cases it is possible to take the axes of co-ordinates, so that 

 #o shall be zero, and also that Y and its differentials shall vanish. In 

 such cases the coefficients take the following simple forms : — 



A = k 2 , 



B = VX u + Ge, 



C = v G u - uqG v - X<j G tt , \- (6). 



D = u,(X v G u -X u G v )-gG } , ' 

 E = a (XvG u — X^Gy), 



