Equations of Hydrodynamics. 443 



disturbances. It is also to be noticed that the equation (3), 

 although it does not contain explicitly accelerative force, was 

 deduced on the principle that the action of the forces is such 

 that changes of position of the surfaces of displacement dp not 

 take place per saltum. Having made these statements, I might 

 content myself with simply referring to the investigation given 

 under Prop. VII,, as I have seen no reason to call in question 

 the exactness of the analysis there employed. Since, however, 

 the proposition is an important one, the investigation will be 

 repeated here in a condensed form. Representing by s any line 

 drawn at a given instant constantly in the directions of the 

 motions of the particles through which it passes, and termi- 

 nating at the point ocyz, and by V the velocity at that point, we 

 shall have 



»£ «•#+*£-*-» f- ' 



Hence, substituting in the equation (3), 

 df d^ 2 _ 



Making, now, the supposition that X is a function of ^ and /, the 

 integration of this equation would give 



f =M t), 



and consequently 



The variation (difr) being supposed to be from point to point of 

 a given surface of displacement, so that, as before, (d-^r) = 0, we 

 have the consequence 



But -j- } being proportional to the velocity V, does not vanish, and 



we must therefore have (ds) = 0. Hence by integration s=c if 

 which equation means that s does not vary in passing from point 

 to point of a given surface of displacement. By taking any other 

 surface of displacement we should similarly have s = c 2) and the 

 difference between the two values of s would be constant. Thus 

 there will be a constant interval between the two surfaces, which 

 cannot be the case unless the trajectory be a straight line and the 

 motion be rectilinear. 



As this result has been reached without employing all the 

 three general differential equations, it cannot be affirmed that 

 rectilinear motion is general and necessary. According to the 

 second of the remarks in art. 9, we may admit that the rectilinear 



