272 
PROFESSOR W. J. MACQUORN RANKINE ON THE 
The following formula is an immediate consequence of the equations (2) : let dx' denote 
an elementary line in any direction, and v! the component velocity of a particle along 
dx’ ; then 
dii! d /w 2 -f« 2 + iv l 
dt dx' y 2 
(11 B) 
In the two previous papers before referred to, a class of stream-lines is described under 
the name of Lissoneoids , whose characteristic property is that two maxima and one 
minimum of the velocity coalesce in one point, at the greatest breadth of the figure 
bounded by the line. The mathematical properties of a lissoneoid are expressed by the 
following set of equations : 
when#=0; letv=0; w— 0 ; ^ 
> 2 +w 2 ) = 0; . 
d 2 
^(w 2 +w 2 +w 2 )= 0; 
(11c) 
and it can be shown that for the last two of these equations the following may be 
substituted in the cases which occur in practice : 
d q u du~ ,, du 1 
tf &S + 2 ^ + 2 ^=°- 
dz^ 
(11 D) 
In order to express the condition that at an indefinitely great distance from the origin 
v and w shall vanish, and u approximate indefinitely to 1, it is necessary that, when 
either x, y, or z increases indefinitely, the functions 4/ and x shall approximate indefi- 
nitely to two functions of y and z only, which may be denoted by 4^ and fulfilling 
the following conditions, 
^0_^Qfko = l ; 
dy dz dz dy 
dx 
0; A >=0; 
dx 
( 12 ) 
that is to say, first, the surfaces represented by 4'o and Xo divide the space into elemen- 
tary streams of equal transverse area ; secondly, these surfaces are plane or cylindrical, 
and parallel to the axis of x ; and thirdly, they are asymptotic to the surfaces repre- 
sented by 4/ and x- Let us now make 
4'=4'o+4d; x=xo+Xi’> 
then the equations (9) take the following form: 
u — 1 _|_f% . dx . ^pj_4i. <b(A _#o d X\ d ^\ d Xo_ d ^\ d Xi 
dy dz dy dz dy dz dz dy dz dy dz dy 
dz dx dz dx dx dz dx dz 
w — , dxo #i dx i _ #q dxi __ #i dx\ ■ 
dx dy dx dy dy dx dy dx ’ j 
(13) 
(14) 
