11G, 117.] KINETIC ENERGY IN SPECIAL CASES. 133 



117. In the case (c) of a solid of revolution, the complete 

 determination of the motion (when no external forces act) has been 

 shewn by Kirchhoff * to be reducible to a matter of quadratures. 



The particular case where the solid moves without rotation 

 about its axis of symmetry, and with this axis always in one plane 

 (i.e. when p = 0, q = 0), has been examined at length by Thomson f 

 and Kirchhoff :. The equations (24) then become 



du, dv A &quot; 



^ = Tt ~ (42). 



dt ^ J 



Let X, Y be the co-ordinates at any instant of the moving origin 

 relatively to axes fixed in space in the plane xy, the direction of X 

 being that of the resultant impulse / of the motion; and let 

 denote the angle (measured in the positive direction) which x 

 makes with X. We have then 



Au=I cos 0, Bv = - /sin 0, r = 6. 



The first two of equations (42), which merely express the fixity of 

 the direction of the impulse in space, are satisfied identically ; the 

 third gives 



or, writing 20 = 



(43), 



the equation of motion of a common pendulum. When S- has been 

 determined so as to satisfy (43) and the initial conditions, X and Y 

 are to be found from the equations 



X = u cos0-v sin 



Y= u sin d + v cos = 

 the second of which gives 



* Crdle, t. 71. Ueber die Bewegiing ernes Eotationkorpers in einer Fliissigkeit. 

 t Thomson and Tait, Natural Philosophy, Art. 332. 



