36 Prof. Challis on the Principles of Hydrodynamics. 



fixed pointj no accelerative force acting : it is required to deter- 

 mine the motion. 



In this instance, for the pui-pose of obtaining an integrable 

 equation, the velocity and changes of density will be supposed 

 veiy small, and powers of the small quantities above the first 

 wUl be rejected. 



Let V= the velocity, and p = \-\-a, o- being veiy small ; and 

 let r be the distance from the fixed jwint, which is taken for 

 oiigin of coordinates. Then 



_Yf .-Yr —Yi 



~ r ' r ' r ' 



d^_du_x dY d^ _ii dY (Pz ^z_ dV 

 dt^ ~ dt~ r' dt' dt' ~ r' dt' df ~ r ' dt' 

 Hence the equations (1.) and (2.) become for this case, 

 ^da- ^ dV _ 



dt dr r 



By eliminating V from these two equations, there will result 



d^. ar _ 2 d'^. or 



1[iF~""~d^- 



This equation is satisfied by the integral, 



ar=f[r—at). 



Or, giving to the function a particular form, 



m . 27r , . 



cr= — sm-^ [r — at). 

 r A 



Keferring now to the principle of the separability of tlie parts 

 of a fluid by an indefinitely thin partition without assignable 

 force, as asserted in Definition II., and the application of that 

 principle in the proof of the law of pressure given in Proposi- 

 tion II., it will appear that in the instance before us the parts 

 of the fluid may at any moment be conceived to be separated by 

 an infinitely thin spherical partition of arbitrary radius. On 

 this account the function / may be taken discontinuoushj , pro- 

 vided the pressures on the opposite sides of the partition be 



equal ; that is, y^ may vaiy per saltmn, but cr may not vaiy per 



saltum. Accordingly the values of the cii'cular function assumed 



above may be taken from r=at to r = at+ ^, and the condensa- 



tion in all the rest of the fluid be supposed to be zero. Thus a 



