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PEOFESSOE W. J. MACQTTOEN EANKINE ON THE 
of s' ; then we have 
dad dy 1 dz' 
ds ' ds' ds 1 
</<p d<p dp 
dx dy dz. 
In short, the stream-lines bear the same relation to the normal surfaces that lines of 
force bear to equipotential surfaces. 
Let the axis of x be taken parallel to the direction of the uniform motion of the par- 
ticles at an indefinitely great distance from the origin of coordinates, near to which the 
solid body is supposed to be situated ; and let the velocity of that uniform current be 
taken as the unit of velocity, so that u, v, and w shall represent the ratios of the three 
components of the velocity of a particle to the velocity at an indefinite distance. Then, 
when either x, y , or z is indefinitely great, we have 
u= 1 ; ; w=0 ; 
and it is evident that the velocity-function must be of the following form, 
<P=x- 1 -£„ ( 7 ) 
in which <p, is a function that vanishes when x , y, or z increases indefinitely. The term 
x gives, by its differentiations, the expression of a uniform straight current, of the velo- 
city 1. The term <p, gives, by its differentiations, the three components of the disturb- 
ance of the velocity from that of the uniform current. Hence, if we suppose the water 
at an indefinite distance from the disturbing solid to be still, and the solid to move 
parallel to the axis of x with the velocity —1, the following coefficients, 
dp, dp, dp , 
dx ’ dy ’ dz ’ 
will represent the components of the velocity of a particle relatively to still water. 
§ 3. Stream-line Surfaces in general. — For some purposes a more convenient way of 
expressing the properties of stream-lines is, to consider the system of stream-lines in a 
steadily moving current of liquid as the intersections of two sets of surfaces called stream- 
line surfaces , represented by the two sets of equations 
^=6; (8) 
where b and c are constants, each of which receives a series of different values. Each 
set of surfaces divides the space in which the current flows into a series of indefinitely 
thin layers ; and the two sets of surfaces divide that space into a series of indefinitely 
slender elementary streams *, which are conceived to be of equal flow. The uniform 
current at an indefinite distance from the disturbing solid being, as before, parallel to x, 
and of the velocity 1, let the transverse area of an elementary stream at an indefinite 
distance be denoted by <r ; the same symbol denotes the volume of the flow in each unit 
of time along that stream, and therefore along every elementary stream. The areas of 
*• Note added June 1871. — Called by Clerk Maxwell “ unit-tubes.” 
