Attractive and Repulsive Forces. 191 



displacement the coordinates of which at the time t are .r, y, z. 

 Then, passing from that point to any other indefinitely near on 

 the same surface, the coordinates of which are x-\-dx, y-\-dy, 

 z + dz, the equation 



- dx + - dy ■\- -- dz = 



r\. A. A- 



will express generally that the motion at each point is in the 

 direction of a normal to the surface, - being a factor which makes 



A 



the left-hand side of this equation a complete differential. Hence, 

 representing that differential by {dy\r), according to the above- 

 stated axiom we shall have, as well as [d-^)=0, also S{d'*Jr)=0, 

 the symbol 8 having reference to change of the surface of dis- 

 placement of the given element by change of the time and of its 

 position. On account of the independence of the symbols of 

 operation S and d, that equation is equivalent to {d .Syjr) —0, 

 But 



and because the variation with respect to time has reference to 

 the change of position of the given element, 



Sx = u8t, Sy = vSf, Sz = ivBt. 

 Hence 



Consequently the differential, with respect to space, of the quan- 

 tity within the brackets [ ] is zero, and by integration 



f ^S-f-f-X(0 (a) 



4. The reasoning thus far has already been given under the 

 head of Proposition VI. in an article "On the Principles of Hy- 

 drodynamics" in the Philosophical Magazine for January 1851, 

 and more recently in pp. 174 and 175 of the 'Principles of 

 Mathematics and Physics.-' It is reproduced here for the pur- 

 pose of drawing an important inference which I had previously 

 overlooked. Before doing so, however, it will be proper to enun- 

 ciate for future use the following general rule respecting the ap- 

 plication of analysis to physical questions. When the funda- 

 mental 4)rinciples of any department of applied mathematics 

 have been expressed in the form of differential equations, the 

 solutions of the equations are coextensive with the physical con- 

 sequences of the principles ; so that there is no such consequence 

 which the solved equations do not embrace, and no positive 



