34 Prof. Challis on Hydrodynamics. 



it to be absolutely necessary for the advancement of physical 

 theory have been fully stated in previous communications. 



For the better understanding of the present argument it will 

 be proper, in the first instance, to advert briefly to several pre- 

 viously established theorems. One of these theorems is, that 

 the third general equation conducts to rectilinear motion by 

 merely supposing that udx + vdy + wdz is an exact differential. 

 As this is an abstract analytical supposition, made without refer- 

 ence to any case of arbitrary disturbance, we may infer from the 

 result that rectilinearity is a general characteristic of the motion 

 of a fluid, in so far as the motion is not determined by arbitrary 

 conditions. To ascertain whether the straight line along which 

 the motion takes place is an axis relative to the contiguous 

 motion, I have supposed that (d ./</>) = udx-\-vdy-\-wdz, and 

 that cj> is a function of z and t, and / a function of x and y, such 



that f= 1, J- = 0, and -J- = 0, where x = 0, and y — 0. This 



supposition is justified by finding that, on introducing it into 

 the two other general equations, an equation is obtained from 

 them which, for points contiguous to the axis, is resolvable into 

 the two following : — 



Of each of these equations I have obtained a particular solution — 

 that is, a solution of explicit form to which the analysis conducts 

 without reference to arbitrary conditions of the motion. (See 

 an Article in the Philosophical Magazine for December 1852.) 

 By the process referred to, the first equation gives 

 2 e2f4 e 3 r 6 



j—l er + p-^22 l 2 . 2 2 . 3 2 ~^~ 

 f being the distance from the axis of z> which is the axis of 

 motion, and e being put for -r— |; and the other equation gives 



for -j-, or the velocity (io 1 ) along the axis, 



clz 



co ! = m sin oil — , 9 — -r cos 2oil + &c, 

 3/ 3fl(/r — 1) 3 



27T 



q being put for %—, p for z — a l t + c, and k being the ratio of a v 

 the velocity of propagation, to a. Also by means of the equation 



