ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 11 



any function of the velocity, the terms due to that resistance in the 

 general formula of motion may be expressed in the form 



where v denotes the velocity and <j>(v) the resistance. But 



xx , yii , zz dv 

 -+ + = ^T = V. 



v v v at 

 The terms due to the resistance reduce, therefore, to 



or, v, 



where / denotes the primitive of the function denoted by <f>. 



Discontinuous Changes of Velocity. Formula (9), which relates to 

 discontinuous changes of velocity, is capable of similar transformations. 

 If we set w z _ AX* + A?/ 2 + A< 2 , 



the formula reduces to 



^2(XA*+YAy+ZA-Jww 8 )^0, (18) 



where X, Y, Z are to be regarded as constant. If g(Pdp) represents 

 the sum of the moments of the impulsive forces, and we regard P as 

 constant, we have 



8 L^(PAp) - 2( Jmw 2 )] ^ 0. (19) 



The expressions affected by S in these formulae have a greater value 

 than they would receive from any other changes of velocity consistent 

 with the constraints of the system. 



Deduction of other Properties of Motion. 



The principles which have been established furnish a convenient 

 point of departure for the demonstration of various properties of 

 motion relating to maxima and minima. We may obtain several 

 such properties by considering how the accelerations of a system, at a 

 given instant, will be modified by changes of the forces or of the 

 constraints to which the system is subject. Let us suppose that the 

 forces X, Y, Z of. a, system receive the increments X', F', Z\ in con- 

 sequence of which, and of certain additional constraints, which do not 

 produce any discontinuity in the velocities, the components of accelera- 

 tion x, y, z receive the increments x', y', %. The expression 



(20) 



will be the greatest possible for any values of x', y', z' consistent with 



the constraints. But this expression may be divided into three parts, 



2[(Z+Z / )+(F+ Y')y+(Z+Zyz-\m(x*+y* +2 2 )], (21) 



2 [Xx f + Yij' + Zz' - m(xx' + yy' + 2*0], (22) 



and 2[Z'^+ Yy' +Z'z'- Jm(o; /2 +f 2 +' 2 )]. (23) 



