ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 13 



whence it appears that ?<(X'x'+ Yy'+Z'z) differs from (Qq') only 

 by quantities which are independent of the relative acceleration due 

 to the additional forces and constraints. It follows that these relative 

 accelerations are such as to make 



(<2^)-2(imu' 2 ) ( 25) 



a maximum. 



It will be observed that the condition which determines these 

 relative accelerations is of precisely the same form as that which 

 determines absolute accelerations. 



An important case is that in which new constraints are added but 

 no new forces. The relative accelerations are determined in this case 

 by the condition that 2 (Jmu' 2 ) is a minimum. In any case of motion, 

 in which finite forces do not act at points, lines or surfaces, we may 

 first calculate the accelerations which would be produced if there were 

 no constraints, and then determine the relative accelerations due to 

 the constraints by the condition that 2(Jmu' 2 ) is a minimum. This 

 is Gauss's principle of least constraint* 



Again, in any case of motion, we may suppose u to denote the 

 acceleration which would be produced by the constraints alone, and u' 

 the relative acceleration produced by the forces ; we then have 



whence, if we write u" for the resultant or actual acceleration, 



2 ( J mu 2 ) + 2 Gmu' 2 ) = 2 ( Jmu" 2 ). 

 Moreover, differentiating (25), we obtain 



whence, since &q', Sx', Sy', Sz' may have values proportional to q', x', 



These relations are similar to those which exist with respect to 

 vis viva and impulsive forces. 



Particular Equations of Motion. 



From the general formula (12), we may easily obtain particular 

 equations which will express the laws of motion in a very general 

 form. 



Let cta^, da) 2 , etc. be infinitesimals (not necessarily complete 

 differentials) the values of which are independent, and by means 



*This principle may be derived very directly from the general formula (6), or vice 

 versa, for S (\mu' z ) may be put in the form 



the variation of which, with the sign changed, is identical with the first member of (6). 



