826 
PROFESSOR 0. REYNOLDS ON CERTAIN DIMENSIONAL 
the vessel, may be considered as a small space, such as a small aperture in the side of 
the vessel or a small hot body within the vessel, the effect of such steady flow will be 
to cause a varying condition throughout the gas. The lines of flow, whether of heat or 
of gas, will diverge from the exceptional space, and the surfaces of equal pressure and 
temperature will be everywhere perpendicular to the lines of flow of matter and heat 
respectively. Except in the case of absolute symmetry, the lines of flow will not be 
straight, nor will the directions of the lines of flow in the immediate region of any 
point focus in a point. 
But in the immediate neighbourhood of any point P, the direction of the lines of 
flow must be such that the directions of the lines of flow parallel to some plane, x y } 
will converge to some point C, while the directions of the lines of flow parallel to the 
perpendicular plane x z will converge to some point C' in CP, which it will be seen is 
taken parallel to the axis of x. 
Whence putting PC —r y and PC'= r : , the surface of equal pressure or temperature 
at P will be a surface perpendicular to x, and having r y and r z as its principal radii of 
curvature in the planes of x y and x z respectively. The simplest cases are those in 
which either the two radii are equal or one is infinite, and these are the cases which 
will for the most part be considered. 
It will at once be seen that at any point within gas in the condition just described, 
p, «, ^+r 2 + w~ and IP+W-f-W~ are functions of r y r~. 
Also, remembering' that the axis of x is taken in the direction of the lines of flow 
7 O 
at P, the point considered, we see that at P, Y, W, v, w, are severally zero, as are also 
r/U d U (FV dm dV dW 
dy ’ dz’ df’ dz~ ’ dx* dx ’ 
while we have 
dm _ 1 dU_ 1 
dy% r v dx ?y 2 
dm _i tfu_ i 
dz~ r, ~dx r? ’ 
civ __u 
dy — V 
dW _U 
AT — r/ 
d 2 Y_1 
dxdy r y dx rf 5 
dmji 
dxdz r, dx r? 
(118) 
Also putting /(pot.) for any function of p and a, — /(pa) and — f(pa) are zero, while 
