MATHEMATICAL THEORY OF STREAM-LINES 
275 
Chaptee II. Summary of Principal Properties of previously known Special Classes of 
Stream-lines. 
§ 5. Stream-lines in tivo Pimensions , especially those with tiro Foci. — Following the 
first of the two methods mentioned in § 3, let the simplest of all possible forms be 
assigned to the function viz. %—z. This form represents the division of the liquid 
mass into an indefinite number of layers of uniform thickness, by a series of plane 
stream-line surfaces parallel to x and to y ; and it involves the supposition that all the 
motions of the particles of liquid take place parallel to the plane of x and y. 
The equations (9) in this case become the following : 
u ~w •=-& w =° (“> 
The equations (11) become the following: 
dv d 2 4 / 
dz dzdx 
du 
dz 
du 
dy 
dx dx~ 
+ 
dy 2 
0 . 
(17) 
The equations (12) and (13) are reduced to the following: 
# 01 . 1 
d, J l .... . (18) 
and therefore %f/ 0 =y, and ; J 
where is a harmonic function in two dimensions ; that is, one fulfilling the condition 
(19) 
The equations (14) become the following: 
u=l+^l;v=-^. .......... ( 20 ) 
ay ax 
The preceding equations show that the stream-line surfaces are cylindrical (in the 
general sense), with generating lines parallel to the axis of z , and that they have asym- 
ptotic planes parallel to the plane of zx. The traces of those asymptotic planes on the 
plane xy are a series of equidistant straight lines parallel to the axis of x, and correspond- 
ing to an arithmetical series of values of b in the equation y=b , being the stream-lines 
of a uniform current in a plane layer of uniform thickness. 
The simplest case of disturbance of such a current by a solid body is that in which 
the disturbance may be represented by a radiating current, diverging from an axis in the 
plane of zx, within the solid body and parallel to z, and converging either towards the 
same axis, or towards a second axis similarly placed ; and this is the mode of production 
of the bifocal stream-lines in two dimensions, or oogenous neo'ids, whose properties are 
investigated in detail in a paper “ On Plane Water-lines,” published in the Philosophical 
mdccclxxi. 2 Q 
