124 Mr. Challis on the Genei-al Eqtiations 



Integrating, -^-^ =f{t\ for the difFerentiatious are relative 

 to X, 7/, z, t being constant. Integrating again, 



'■?=/(«)'■ + ?(') 

 ... , =j(t) + ^i 



The velocity = V^-f )' + (^0 + GD' = 2' «»- 

 i{ q z= the velocity. 



The value of <p contains tvi^o arbitrary functions, as it ought; 

 also it is such that ((Z<p) is a complete differential of a-, y, and 

 z. Nothing appears in the mode of obtaining the above inte- 

 gral to forbid our saying that it is the proper general integral 

 of the differential equation. The supposition x^ -^^ ■ip ■\- •:? = r% 

 by no means limits its generality. We are rather taught 

 something about the general character of the motion by the 

 possibility of obtaining $ in terms of a single variable. Plainly 

 r is the distance of the point under consideration from the 

 origin of coordinates ; and the inference to be drawn is, that 

 in general a particle is moving in such a manner that its velo- 

 city varies inversely as the square of its distance from some 

 point, and its motion is directed either from or towards this 

 point. I say some ■point, because the origin of coordinates is 

 perfectly arbitrary. The generality of the infei'ence is legiti- 

 mately deduced both from this circumstance, and because as 

 no supposition was made about the manner of putting the 

 fluid in motion in the investigation of the differential equa- 

 tions, so none has been made in the foregoing reasoning 

 founded on them. All this will be clearly understood by 

 conceiving a sphere of the fluid to be inclosed in an envelope 

 capable of expansion, and a small spherical ball of solid matter 

 to be introduced into it, and placed concentric with it. Let 

 r = the radius of the ball, 11 = that of the sphere before the 

 introduction of the ball, and 8 its increment afterwards. 



Then tt R^ 8 = — — , r being very small, 



8=1^:1 



8 will also represent the translation from its original position 

 of a particle at any distance R from the centre. Consequently 

 if the particles be conceived to change their positions b}' a 

 uniform motion of translation ui the directions of the radii of the 



sphere 



