MATHEMATICAL THEOEY OE STEEAM-LINES. 
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and all the terms in those expressions, except the 1 in the value of u, represent velocities 
of disturbance produced in a still mass of liquid by the motion of a solid parallel to x 
with the velocity —1. 
The form of the disturbing solid may be represented by an equation of one or other 
of the following forms : 
4=0; z =0; F(4, x )=0 (15) 
In the problems described in the sequel, the first of those expressions is supposed to 
be used for the figure of the surface of the disturbing solid, viz. 4 = 0; so that 4 —b 
with an unlimited series of increasing values of b, expresses the figures of a series of 
stream-line surfaces lying between successive layers of liquid that enclose the solid within 
them, like concentric tubes. A series of negative values being given to b, correspond to 
a set of internal stream-lines , which represent currents circulating inside the disturbing 
solid. In the present investigation, the external stream-lines alone will be considered. 
The equation %=c, with a series of values of e, represents a series of stream -line surfaces 
which meet the surface of the solid (-^ = 0) edgewise, intersect the surfaces denoted by 
■^ — b, and subdivide the previously mentioned layers of liquid into elementary streams 
of equal flow. 
Two alternative modes of proceeding may be followed in the proposing and solution 
of problems as to the figures of the stream-line surfaces'*. One is as follows : a form is 
assumed for the function satisfying equations (13) and (12) ; and thence are deduced, 
by means of the equations (11), corresponding forms of the function \f/, denoting figures 
of the disturbing solid and of its enclosing stream-line surfaces ; and this is the method 
which has been followed in previous researches, and which will be followed as regards 
the quadrifocal stream-lines or cycnogenous neo'ids specially treated of in this paper. 
The other mode of proceeding is to assume for the function ^ a form satisfying 
equations (13) and (12), and denoting certain figures of the disturbing solid, and of the 
enclosing stream-line surfaces, and thence to deduce by the aid of the equations (11) the 
corresponding form and values of the function %, and the figures of the elementary 
streams. 
From the form of the equations of condition (11) it is easily seen that, if with a given 
assumed form of either of the functions %, there are several forms of the other function 
which satisfy those equations, then every form obtained by addition or subtraction of 
those forms will satisfy them also. In symbols, let % be a given form of one of the 
functions, and any one out of several forms of the other function which, taken along 
with ■£, satisfy the equations ; then any function which can be expressed by X . ^ will 
satisfy them also. 
§ 4. Graphic Construction of Stream-lines. — Let one side of a piece of paper be taken 
to represent one of the surfaces whose equation is %=c. Then the stream-lines which 
* Note added in June 1871. — It is to be observed that those methods are tentative only ; that is to say, they 
may fail when tried, and repeated trials may be necessary before a solution is obtained. 
