344 ON THE MOTION OP 



Md%-j-SX,Dm = o, 



Md\^SY,Dm = o, 



Md% + SZ,Dm = o; 



U-hS(Z,ij, — ¥ l z,)JDm = o, 



V + S(X,z,—Z;x,)Dm = o, 



W-^s{Y,x,—X l y,)Dm. = o. 



of which the first three may, by means of equations (10) 

 and (11), be made to assume the following more usual form 



(17) 



Md'i -i-SXDm = o, 



Md*£'-hSX'Dm = o, 

 MdT + SX"Dm = o. 



■- 



But if the body, as in the problem I have proposed to examine, 

 is forced to roll or slide on a given surface, the above varia- 

 tions are no longer independent, and we must ascertain the 

 influence which the progressive and rotatory motions have 

 upon each other ; or to give this question the geometrical form 

 which the nature of variations seems essentially to require, it 

 is necessary to determine the geometrical relations which a 

 given limitation of position will occasion among the elementary 

 changes of those magnitudes on which the station and the as- 

 pect of the body depend. For this purpose, let K=o repre- 

 sent the equation of the given supporting surface referred to 

 the axes fixed in space, and let K, = o be the equation of the 

 surface of the given oscillating body referred to its own axes. 

 Let L, L', L", L„ M„ N„ represent the cosines of the space 

 and body axe-angles made by the normal common to both 

 surfaces at the point of variable contact P, for whose space and 

 body coordinates we may employ the symbols x, x 1 , x", 

 x„y„z„ so as to make the formulas (3) applicable to these 

 coordinates, recollecting only that x„ y„ z, are now variable 

 quantities. Then because the normal is at right angles to the 



