336 ON THE MOTION OF 



pendent of the direction of the axes in space, it may be pre- 

 sented in either of these forms, ( 1 ) 



SDm[(d*x -+- X)dx -j- {dTx' + X')dx' -+- {d'x"-{- X")dx"] = o 

 SDmftd'x, -+- X)hx, -+- {dry, -f- Y,)dy, + {dTz t + Z )&, ] == o 



where it must he carefully recollected, that in consequence 

 of the motion of the body axes, the variations and accelera- 

 tions in the latter formula, as well as the velocities dx„ dy„ dz,. 

 d£„ dy;„ tf£,, belong to the class of incomplete differentials. 



In these equations the variations are of different values for 

 different elements of the body, or in other words are functions 

 of the coordinates of Dm. It is evident, however, that before 

 this formula can be employed, these variations will in general 

 require to he reduced to other variations common to all the 

 elements, so that, in the language of the calculus, they may 

 be passed from under the sign S. The manner of effecting 

 this, by a general method for all constitutions of matter and 

 for all conditions of motion, must have been a problem of no 

 ordinary difficulty. Mathematicians however have succeeded 

 in this transformation by several processes equally remarka- 

 ble, each of them terminating in an equation of the form 



Lda + Mdp-hMy-t-L'dZ,-t-M'd[i-hN'dv = o. 



In all these transformations, da, d(3, by are the progressive 

 variations common to all the particles in the direction either 

 of the body-axes or the axes in space ; but with respect to the 

 variations b'K, d(i, dv, there exists between these methods an 

 essential difference which deserves to be noticed. To render 

 this distinction the clearer, it is necessary to observe that the 

 absolute position of a body in space involves two considera- 

 tions: 1st, the position in space of some fixed point O, of the 

 body, which may be denominated the station of the body ; and 

 2dly, that part of the position which depends only upon the 

 direction of the body-axes, and which, for the sake of brevity, 

 may be called the aspect of the body. A body therefore may 



