Prof. Challis on the Principles of Hydrodynamics. 95 



but vary from one point to another. We have now to show 

 that motion composed of the normal motions in the manner just 

 stated^ may satisfy this condition respecting the directions of the 

 lines of motion. 



In the case of free motion, we have 



Velocity parallel to the axis =mf sin —- {zl^ Kat ■]- c), 



A, 



Velocity transverse to the axis = — — • ■— cos — {2 + Kut + c) . 



Letf=l—er^, r being very small, and let the direction of the 

 motion, which, by reason of the assumed value of f, must pass 

 through the axis of z, make an angle a with that axis. Then 



since ^ = —2er. and eX^=4, 

 dr 



4r 27r , _ ^ . 



«= — r-COt— - (Z-\-Kat-i-C). 

 TTA/ A. 



Now this equation is satisfied independently of the value of r, if 

 \=k{z + Kat + c) and r=ot{z^Kat-\-c)j the constant k being 

 determined by the equation 



1= —5 cot-i— 



TTK k 



The directions of the motion at all points of a given section per- 

 pendicular to the axis will thus pass through a point of the axis 

 whose distance from the section is the arbitrary quantity 2- + Kat + c. 

 By means of the above value of X and the arbitrary quantity w, 

 the circumstances of the motion in a rectilinear tube whose 

 transverse section is everywhere a circle, may be satisfied at a 

 given position in successive instants, or at successive positions 

 at a given instant. It is worthy of remark that this reasoning 

 has required the equation eX^=4. 



If the motion be directed to focal lines, the value of/ to be 

 used is 



/= 1 — 2e(a7 cos ^ + y sin 6fj 



6 being an arbitrary angle, and a? and y very small. For it was 

 previously shown that the difierential equation which gives / is 

 satisfied by the equation f=2> Ve(x(io^6 + y %mO), whatever 

 be 6. The components of the motion parallel to the axes of 

 coordinates being in this case 



/# 4/" df 



^^P *^' *^ 



the directions of the resultants at all points of a given transverse 

 section may be made to pass through two focal lines in given 

 positions by means of the additional arbitrary quantity 6, In 



