358 ON THE MOTION OF 



which, by means of the equations of the surfaces, will become 



*; ~~ + W ¥7 r:- ) ' 

 k — *■■"" V? -7" ^ "^ <?v ' 



If now we substitute the first of these two values in the first 

 triplet of equations (30) and then substitute the values of 

 x, x', x" thus transformed in the first triplet of equations (2), 

 x„ y,, z, may be determined by quadratics in terms of £ /? ^, £ 

 and the aspect of the body. By a process altogether similar, 

 x, x', x" may be obtained in terms of £, £', £". At the same 

 time it ought to be observed, that whatever be the nature of 

 the surfaces, if from the seven equations K = o, K l = o, 

 either of the triplets (2), and any two of the three equations 

 of contact (20) (the three being in fact equivalent to two in 

 consequence of the condition L % -t-i/ r -\-L*= Lf-^M^N, 3 ) 

 we eliminate the space and body coordinates of the point of 

 contact, there will remain an equation of condition between 

 the elements of the station and the aspect of the body, of 

 which equation (22) is in all cases the differential. 



As the angular velocities p, q, r are functions of the nine 

 cosines and their differentials, and as these are connected by 

 six equations of condition and variously expressible in functions 

 of the three elements of the aspect of the body, it follows that 

 the six equations of motion will by the above mentioned sub- 

 stitutions involve, beside the time, the six elements of tbe 

 position of the body. 



If we substitute in place of L,, M, N, in the equations of 

 motion their values as given by equations (29) and employ at 

 the same time the abridgments, 



Q — B, = J. J —C,=B^ B,—A, = C 



a 5 



