Prof. Challis on the Principles of Hydrodynamics. 97 



proposition we have only to substitute Ka for a', and we thus have 

 generally 



Y-X{m,t)=Ka(T=-j^(l){2-Kaf-{-c). 



Corollary. The effective aecelerative force of the fluid in the 

 direction of the axis of the tube may be inferred from the fore- 

 going results. By the effective aecelerative force of the fluid is 

 to be understood that part of the actual effective aecelerative 

 force which is independent of the variations of the arbitrary 



quantities %(wi_, t) and .^ . , or which is obtained by supposing 



these quantities constant through an indefinitely small space and 

 for an indefinitely small time. Thus we have 



dY fjLKa , dY 



-j7 = — ■^j-r^<p\z—Kat + c)-=—Ka-j-. 

 dt ^jr^z) ^ ^ ' dz 



But it was found by the reasoning of Proposition XII.^ that if 

 the variation of m be omitted, to the first approximation 



Hence 



dY__ da 



dz "" dz' 



dcr dJl 

 """'dz^ dt -"• 



It thus appears that the effective aecelerative force of the fluid 

 in a slender rigid tube is greater than that of fluid in free motion 

 in the ratio of k^ to 1 . 



The following problem has been selected for solution, as being 

 illustrative of various parts of the foregoing analytical theory. 



Problem. The fluid is disturbed in such a manner that the 

 velocity and condensation are everywhere and at all times func- 

 tions of the distance from a fixed centre, and the velocity tends 

 to or from the centre : it is required to determine the motion. 



It will plainly be sufficient to consider the motion in a slender 

 tube bounded by planes passing through the centre. The 

 boundaries of the tube may be conceived to be rigid, and con- 

 sequently, from what has been proved, the differential equations 

 to be employed are the following : 



"""ds^ dt -^' 

 da- du dv dw _ 

 dt doc dy dz ~ ' 



It is now permitted to introduce into these equations the con- 

 dition that V and a are functions of R the distance from the 



