Prof. Challis on some Theorems in Hydrodynamics. 339 



Now introducing the condition that the motion is rectilinear, 

 obtained, as above stated, in a general manner, dR will be the 

 increment of space, dato tempore, along a line of motion. Also 

 ^R = 6?R/ from the geometrical relation between the principal 

 radii of curvature. Hence integrating the above equation along 

 the line of motion, we have 



v ~ HP 



As this equation has been arrived at without supposing any case 

 of motion, it expresses a general law of the motion of an incom- 

 pressible fluid. I proceed now to an analogous investigation for 

 a compressible fluid. 



It will be required, first, to establish a general theorem respect-* 

 ing the relation between velocity and density in a state of pro- 

 pagation. In accordance with the condition that the lines of 

 motion are normals to a continuous surface, let us suppose the 

 fluid to be bounded by plane faces passing through focal lines, 

 so as to be contained in a slender tube, whose length is very 

 small and transverse section quadrilateral. Let P, Q, R be 

 three positions on the axis of the tube, separated by equal inter- 

 vals. Then it will be required to solve the following general 

 problem relating to propagation : viz. to express the rate at 

 which the excess of fluid in the space between Q and R above 

 that which would exist in the same space in the quiescent state 

 of the fluid, becomes the same as the excess in the space between 

 P and Q. 



Let V, p be the mean velocity and density of the fluid which 

 in the small time 8t passes the section at Q, and V, p' the same 

 quantities relative to the section at R. Let the magnitude of 

 the section at Q be m, and of that at R be m', and the interval 

 between them be Bz. Then the increment of matter in the time 

 St in the space between Q and R is 



VpmSt-Vp'm'St. 



Let this be equal to the excess of the matter in the space between 

 P and Q due to the state of motion, above that in the space 

 between Q and R, at the commencement of the small interval St. 

 The expression for this excess, neglecting small quantities of the 

 second order, is 



(p ■— 1 )mSz — (p* — 1 ) m'Bz. 

 Hence we have the equation 



d . Vpm ___ d . (p — l)m 8z 

 dz dz 8t ' 



Sz 

 which gives the required expression for the rate k-. 



