MATHEMATICAL THEORY OE STREAM-LINES. 
271 
the three sections of an elementary stream, made at a given point by three planes parallel 
to the three coordinate planes respectively, have the following values : 
parallel to yz, : 
dy dz dz dy 
and symmetrical expressions for those parallel to zx and to xy respectively. 
The three components of the velocity of an elementary stream at a given point are to 
be found by dividing the volume of flow by the areas of those three sections respectively ; 
hence those components are as follows : — 
<fy'dx_dyb'dx_dl 
dy dz dz dy dx ' ' 
(and symmetrical expressions for v and w). 
The third member of the equation is introduced in order to show the relations between 
the stream-line functions \J/ and and the velocity-function <p. 
It is easily ascertained that the preceding values of u, v, and w fulfil the condition of 
constant density (equation 1); also that the surfaces of equal action (<p = a) cut the 
stream-line surfaces at right angles, as expressed by the following equations : 
d-\> d<p d*\> d§ d-\i d<$ ^ ^ 
dx dx' dy dy ' dz dz ’ 
dx dx' dy dy * dz dz 
( 10 ) 
The conditions expressed by the three equations (2) take in the present instance the 
following form : 
„ dv dw 
dz dy 
dfy _ d*x , _ d^x 
dy dxdy ' dz dzdx 
_ d±/d?x d?x\ 
~ dx\dy 2 ~^ dz*J~r 
. dx / d z ty rf a 4>\ _ dx _ d*ty __ dx _ 
' dx l dy 2 ' dz 2 J dy dxdy dz dzdx ' 
0=^— ^=( expression formed by symmetry) ; 
0=^—^— (expression formed by symmetry). 
( 11 ) 
The preceding set of three equations show the whole conditions which the functions 
and x must fulfil, in order that they may represent stream-line surfaces. 
In finding the point in a stream line where a given function F is a maximum, the 
condition to be fulfilled is 
did 
dt 
( U i+ V Ty+ W l) F==0 - 
(11 i) 
