Prof. Challis on the Principles of Hydrodynamics. 4!47 



D being an arbitrary function of x, y and ~ . It cannot, there- 

 fore, be argued, as in Prop. VII., that i/r is a function of s and^, 

 and by consequence that the motion is rectiUnear. Hence the 

 integrabiUty of udx + vdy + wdz for small values of u, v, w is 

 consistent with curvilinear motion, and may be satisfied by the 

 motion at any distance from the axis of rectilinear motion. 



To cany on the investigation of the law of action of the parts 

 of the fluid on each other to the first order of approximation, I 

 shall continue to use the same expressions for the velocities as 

 in the general case; but in consequence of what has just been 

 proved, these expressions will not now be restricted to points 

 contiguous to the axis of rectilinear motion. This extension of 

 their application will be justified if it leads to no contradictory 

 results. Thus we shall have 



"=•^1' ^='^1' "^=^§' . . . . w 



at any point whose coordinates are x, y, z, and at any time t, 

 f being a function of x and y only, and ^ a function of z and t 



only. 



The equation which gives the condensation <t to the first order 

 of approximation is 



in which ¥{t) must be made to vanish in order that the reasoning 

 may be conducted independently of any arbitrary circumstances. 

 Consequently, after determining (j> and/, the value of a is given 

 by the equation 



"'-'+/f=o W 



The equation (S) to the first approximation becomes 



Now as <j} is independent of x and //, it has the same value at 

 all points of any plane perpendicular to the axis of z, and there- 

 fore the same value as at the point of intersection of this plane 

 with the axis. But we have seen that for points on the axis the 

 following equation is true to the first approximation, viz. 



*'*-^'3 + f=o w 



Hence substituting in (X) from (/i), and striking out the common 

 factor (j), we obtain 



d^f dH bH ^ 



