106 Prof. Challis oz Rectilinear Fluid Motion. 
dv _dw_V cos B cos? ¢ — cos? ¥ =: Ojansi{ ¥ah 
dz dy raat — cosy “cos? b — cos? B ay ) i 
dt bata 
w du igeagnt foto Sal ga cos pli died 0. (VL) 
dx dz R — cosa / cos?c — cos? y 
The general verification of these equations requires R to be 
infinitely great, that is, the motion to be rectilinear. Hence, 
whenud«+vdy-+ wd zis an exact differential of a function 
of three independent variables, not only is the motion rectili- 
near, but the velocity must also be the same at all points of 
the same surface of displacement, ‘This accords with what I 
have otherwise proved. 
When the motion is parallel to the plane of x y, we have 
7 T T 
Bas Oh = Oitmees gh ech Alia wih vias 
du dv 
and the complete expression for —- — —— becomes 
dy ~ da 
dv sap sin « + —— cos 2 
ay POR WE + R Ae 
It is possible that this quantity may vanish when neither of 
the equations (I.) and (IV.) is satisfied. ‘This will happen 
when the motion is in concentric circles about an axis, in 
which case we have 
OVI Bee, Ve So dV 
Diemariaa:, ee. Mala ae d 
HS abY Se ow aR AS dB eh. widy PreRe 
GN 
Hence (Gt Rr) 228 2 == 10} 
Therefore, = oe L = 0, and by integration V = —_ 
This is the singular case of curvilinear motion which I 
have already treated of in a different manner, 
Cambridge Observatory, Dec. 17, 1842. 
Postscript, Jan. 5, 1843.—The foregoing discussion super- 
sedes the necessity of answering at any great length Mr. 
Stokes’s reply in the January Number. All that is urged by 
Mr. Stokes in p. 55, is sufficiently answered by the reason- 
ing which has conducted me to the equations (9.) and (10.), 
and by the consequences resulting from these equations. ‘The 
principle on which the independence of the variations d x, dy, 
