DE. W. J. MACQUOEN EANKINE ON PLANE WATEE-LINES. 
371 
the solid as fixed, and the infinitely distant particles of the fluid as moving uniformly 
with an equal speed in the contrary direction. Throughout the present paper, the solid 
will be supposed to move along the axis of x ; so that v will represent the transverse com- 
ponent of the velocity of a particle of liquid on either supposition. The longitudinal 
component of the velocity of a liquid particle relatively to the solid will be denoted by u ; 
and when that particle is at an infinite distance from the solid, by c; so that when the 
infinitely distant part of the liquid is regarded as fixed, the solid is to be conceived as 
moving with the velocity — c; and the longitudinal component of the velocity of a 
liquid particle relatively to the indefinitely distant part of the liquid will be denoted by 
u—c. ■ 
It is convenient to regard the function U as equivalent to an expression of the 
following kind, 
b V=bc, (7) 
c being the uniform velocity of flow at an infinite distance, and b what the value of y 
would be for the water-line under consideration if the solid were removed ; in which 
case that line would become a straight line parallel to the axis of x. This enables us 
to substitute for equations (5) and (6) the following, in which proportionate velocities 
only are considered : — 
u db v db , Q . 
r. dn ’ c d.x’ ' ' 
dfb dfb_ 
dy*+dx *— U ‘ 
( 9 ) 
(4.) General Characteristics of Water-Line Functions . — Since at an infinite distance 
from the solid body we have u=c, v=0, it follows that, if the origin of coordinates be 
taken in or near the solid body, b must be a function of such a kind that, when either 
x=co , or y= oo , 
b-y. 
Hence in a great number of cases that function is of the form 
b=y+F(x,y); (10) 
where F is a function which either vanishes or becomes constant when x or y increases 
indefinitely. 
It is plain that when the function b takes this form, the term F is the function for 
the motions of the liquid particles relatively to still water ; that is to say, 
u—c db - dF ' V db dF 
c ~ dy~ dy * c ~ dx dx ’ (H) 
and also that the term F fulfils the equation 
d?F dfF 
dy 2 dx 2 
( 12 ) 
