278 
PEOFESSOE W. J. MACQTJOEN EANKINE ON THE 
As it is sufficient to determine the traces of the stream-line surfaces of revolution in 
any one of those planes, we may take the plane of xy , for which z = 0 ; and then we have 
the following values : 
d X_ 
d X_ 
o ; dx — o ; dy~®’ 
d x_\. d *X_ (P X__ {] . 
dz y ’ dy 2 dz 2 ’ 
fx = _l. 
dydz y 2 
1 
J 
(27a) 
When the preceding substitutions are made in the equations (9) and (11), they are 
converted into the following: 
The equations (9) become 
d\ 1/ d\ \> A 
* =0; 
(28) 
and the equations (11) become 
dv d 2 4> 
0 = 
du d 2 -\> 
dz 
ydzdx ’ 
0=^= 
dz ydydz 
(and therefore ^=0); 
„ du dv d-]) 1 / d 2 ']/ d 2 4/\ 
dy dx y 2 dy y y dx 2 ' dy 2 J ' 
(29) 
The same substitutions being made in the equations (12) give the following results 
^. = 1; and therefore ^o=— (30) 
ydy Y0 2 v ’ 
This last equation shows that the stream-line surfaces which represent a uniform 
current, and are asymptotes to the actual disturbed stream-line surfaces, are a series of 
concentric circular cylinders described about the axis of x, the half squares of whose 
radii are in arithmetical progression. The traces of such a series of cylindrical surfaces 
are represented in fig. 3 by the straight lines parallel to the axis O X. 
The simplest case of the motion of disturbance produced by a solid of revolution 
whose axis is the axis of x , is represented by a current diverging symmetrically in all 
directions from a focus in that axis, and afterwards converging towards another such 
fcous. The stream-line surfaces of revolution about that axis which represent a diver- 
ging or converging current alone, as the case may be, are obviously a series of cones with 
the focus for their common apex, cutting a spherical surface described about that apex 
into equal zones. The function which represents the traces on the plane of xy of such 
a series of conical stream-line surfaces is the following : 
, , k 2 x—a kr cosfl 
Y, = ± V V {( ff - fl ) 2 + y 9 }” ~ 2 ; 
k 2 ■ 
in which a denotes the distance of the focus from the origin of coordinates and ±— is 
dm! 
a parameter, to be used with the positive sign for convergence and with the negative 
sign for divergence. 
