Conditions of Rectilinear Fluid Motion. 105 
Before quitting the subject I am desirous of adverting to 
the demonstration I gave in the Philosophical Magazine for last 
August (S. 3. vol. 21. p. 102.), which Mr. Stokes has objected 
to because it takes no account of the curvature of the lines of 
motion. Notwithstanding this omission, the proof is perfectly 
valid as far as it goes, and requires to make it complete only 
the enunciation of the following principle, which I suppose 
will be conceded, viz. that in fluid motion there is no general 
relation between the curvature of a surface of displacement 
and the curvature of a line of motion. This being admitted, 
when wd a + vdy + wdz is the differential of a function of 
three independent variables, the equations 
du dv du _dwdv_ dw 
dy dx dz” dz dz dy 
must be satisfied both when the curvature of the surface of 
displacement is considered apart from the curvature of the 
line of motion, and when the latter is considered apart from 
the former. The proof in the August Number takes account 
only of the curvature of the surface of displacement. Sup- 
posing «, 8, y to be theangles which the direction of the velocity 
V at the point # y z makes with the axes of x, y, 2 respectively, 
it was shown that the equations to be satisfied (somewhat dif. 
ferently expressed), are, 
dur die aN dV 
ge Baume ea pipe OP On ge NY 
dv dw dV dV 
PAF ig He eg gee ee sf ste EL,) 
dw du _ dav dV 
anda oie ae . . (III.) 
and that the only condition required for satisfying them is, 
that the surface of displacement be a surface of equal velocity. 
Let now R be the radius of curvature of the line of motion 
at the point 2 y x, and let a, b,c be the angles which the 
plane of curvature makes with the axes of a y z respectively. 
Then, abstracting from the curvature of the surfaces of dis- 
placement, that is, supposing these surfaces to be plane for a 
given small element of the fluid, it may without difficulty be 
shown that the equations to be satisfied are, 
du _dv_V_ [cosa ¥ cos?b — cos? B 
Sa ; Bap te Ma a 
mae OR cos Bo aaxa oats f aaiar*) 
