6 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



form the subject of inquiry. Its position in the immediate future 

 is naturally specified by 

 x+xdt + J#cft 2 + etc., y + ydt + $ydt 2 +etc., z+zdt + %zdt z +etc., 



and we may regard the variations of these expressions as corre- 

 sponding to the Sx, Sy, Sz of the statical problem. It is evidently 

 sufficient to take account of the first term of these expressions of 

 which the variation is not zero. Now, x, y, z, as has already been 

 said, are to be regarded as constant. With respect to the terms 

 containing x, y, z, two cases are to be distinguished, according as 

 there is, or is not, a finite change of velocity at the instant considered. 

 Let us first consider the most important case, in which there is no 

 discontinuous change of velocity. In this case, x, y, z are not to be 

 regarded as variable (by <5), and the variations of the above ex- 

 pressions are represented by 



J&jcft 2 , $8ydt 2 , $32 dt 2 , 



which are, therefore, to be substituted for Sx, Sy, Sz in the general 

 formula of equilibrium (8) to adapt it to the conditions of a dynamical 

 problem. By this substitution (in which the common factor $dt 2 

 may of course be omitted), and the addition of the terms expressing 

 the reaction against acceleration, we obtain formula (6). 



But if the circumstances are such that there is (or may be) a 

 discontinuity in the values of x, y, z at the instant considered, it is 

 necessary to distinguish the values of these expressions before and 

 after the abrupt change. For this purpose, we may apply x, y, z 

 to the original values, and denote the changed values by x+Ax, 

 y-\-Ay, z-\-Az. The value of # at a time very shortly subsequent 

 to the instant considered, will be expressed by x-\-(x+ Ax)dt-{-ekc., in 

 which we may regard Ax as subject to the variation denoted by S. 

 The variation of the expression is therefore S Ax dt. Instead of mx, 

 which expresses the reaction against acceleration, we need in the 

 present case in Ax to express the reaction against the abrupt change 

 of velocity. A reaction against such a change of velocity is, of 

 course, to be regarded as infinite in intensity in comparison with 

 reactions due to acceleration, and ordinary forces (such as cause 

 acceleration) may be neglected in comparison. If, however, we 

 conceive of the system as acted on by impulsive forces (i.e., such 

 as have no finite duration, but are capable of producing finite changes 

 of velocity, and are measured numerically by the discontinuities of 

 velocity which they produce in the unit of mass), these forces should 

 be combined with the reactions due to the discontinuities of velocity 

 in the general formula which determines these discontinuities. If 

 the impulsive forces are specified by X, Y, Z, the formula will be 



[(X-mAx)SAx+(Y-'niAy)SAy+(Z--mAz)SAz)]^0. (9) 



