20(> Prof. ChalUs on the Hydro dynamical Theory of 

 But bv the equation (e), 



dt ■ \ 



d.ylr 

 Hence. substitutiD2: X'^ for ^ , 

 ds 



Introduciuo: now the condition that X is a function of-yjr and f, 

 this equation must admit of an integral of the form y!r=f{s, t), 

 in which s is, by definition, the length of a trajectory of surfaces 

 of displacement. Now we may suppose that at the time / one 

 extremity of the trajectory is on a certain given surface, and the 

 other on the undetermined surface which passes through the point 

 whose coordinates at the time t are x, y, z. Accordingly the 

 above value of -^ may be taken to be general as to space for the 

 time ty so that its variation with respect to space gives for any 

 position : — 



25. Since ^d\V^ is the variation of the function -^ in passing 

 at the given time from the point o:yz to any contiguous point, 

 we may inquire, as in the Calculus of Variations, under what 

 conditions that function may have a maximum or minimum 



value. Let. therefore, 'cxV =0. TheUj since " '"/"' % which is 



ds 



d-& Y 

 equal to -j- or — ; may be supposed not to vanish, the sole con- 

 dition is that (8s) =0. This result signifies that in the case of 

 maximum or minimum, contiguous trajectories, intercepted be- 

 tween two surfaces of displacement separated by an arbitrary 

 interval, are of the same length. But this cannot be the case 

 unless the trajectories are straight lines and the motion conse- 

 quently rectilinear. "We have thus been led to an indication of 

 rectilinear motion by arguing solely from the general equation 

 (e), which was derived from the principle of geometrical conti- 

 nuity. It is to be observed that this inference has been drewn 

 antecedently to any supposition as to the mode of putting the 

 fluid in motion, and that it rests on the abstract assumption that 

 there is a certain form of the function 'yjr which gives it a maxi- 

 mum or minimum value. Hence, if it be urged that the indi- 

 cated rectilinear motion is such as might take place if the velo- 

 city were a function of the distance fiom a centre or from a fixed 

 plane, the reply is that this interpretation is inadmissible be- 



