342 Prof. Cballis on some Theorems in Hydrodynamics. 



longitudinal velocity is ^-, and the transverse velocities are <)>-£ 

 and <t>4~, it follows that the lines of motion drawn at any instant 



in the directions of the motions of the particles through which 

 they pass, are continually varying their positions. This varia- 

 bility is due to the transverse velocities, which, as well as the 

 transverse accelerative forces, are indefinitely small very near the 

 axis. Hence by impressing other indefinitely small transverse 

 forces, we may alter ad libitum the positions of the lines of 

 motion, and even give them fixed positions in space. This, in 

 fact, is done by constraining the fluid to move in rigid tubes of 

 small and arbitrary transverse section, the boundary of the tube 

 being supposed at all points to be inclined at indefinitely small 

 angles to the axis of z. In case the motion be not symmetrical 

 about an axis, the value of / to be employed is/= cos (gx + hy), 

 g z + W- being equal to 4e, and x and y being taken very small. 

 Now these impressed forces, being indefinitely small, do not 

 sensibly alter the total velocity V ; and being transverse, they do 

 not change the rate of propagation. But being impressed in all 

 directions about the axis, they may sensibly alter the relation of 

 the velocity to the condensation. Conceiving, therefore, the 

 fluid to be constrained to move in a straight prismatic tube, we 

 shall have, as in free motion, 



V4/7T , , v 



=msin— \z—K,at-\-c). 



The relation between the velocity and the condensation is to 

 be deduced from the general theorem respecting propagated 

 motion in straight slender tubes, obtained in the previous part 

 of this paper, according to which 



w, + ft 



In the present case <f>(t)=0, a'=fca, and consequently Y=/cao; 

 Thus the motion in the prismatic tube is defined by the equations 



V=tffl<r=msin— (z—tcat + c), 



A. 



It is further to be remarked, that the value of V is not restricted 

 to the above form. For by the principle that the parts of the 

 fluid may be separated by a thin partition transverse to the tube 

 without assignable force, the velocity and condensation on the 

 two sides of the partition being the same, we may suppose the 

 quantities /x., X, and c to vary in an arbitrary manner from ele- 

 ment to element along the line of motion. Consequently we 



