MATHEMATICAL THEOEY OF STEEAM-LINES. 
285 
Equation (45) may be used to solve the following problem : — Given the midship half- 
breadth y 0 , the excentricities of the two pairs of foci a. a', and the inner parameter /r ; 
to find the outer parameter k 12 . 
Chapter IV. Dynamical Propositions as to Stream-line Surfaces. 
§ 12. Resultant Momentum. — The resultant momentum, parallel to x, of any part of 
a given elementary stream is equal to that of an undisturbed part of the same stream 
whose length, projected on the axis of x, is the same. For let <r 0 be the sectional area 
of an undisturbed part of such a stream and 1 its velocity ; then <r 0 dx is the momentum 
of an elementary part of its length. 
Let dx also be the projection on the axis of x of an elementary part of the same 
stream, when disturbed, a the sectional area of that part on a plane normal to x, and u 
its component velocity parallel to x ; then its component momentum parallel to x is 
uadx. But iu 7 is the volume of flow along the elementary stream, which is uniform 
and = <7 0 ; therefore 
uadx—aflx ; 
so that the component momentum parallel to x of any part of an elementary stream is 
simply 
^o("G ^i) > 
in which x 2 and x x are the values of x for its two ends. Consider now an elementary 
stream of indefinitely great length, so that its two ends lie in one straight line parallel 
to x , and are at so great a distance from the disturbing solid that its action on the par- 
ticles at those ends vanishes. The resultant momentum of that stream is the same as 
if it were undisturbed ; and such being the case for every elementary stream, is the case 
for the whole mass of liquid. This conclusion is expressed by the following equations, in 
which the integrations extend throughout the whole liquid mass outside the surface of the 
disturbing solid 
S$(u-l)dxdydz=0; | 
Jjj vdx dydz=0; JJj w dx dydz=0.j 
The resultant momentum ))) udx dy dz is that of the liquid relatively to the solid, con- 
sidered as fixed. 
If we next consider the centre of mass of the liquid as fixed, the resultant momentum 
of the liquid becomes 
f\f(u—l)dx dy dz = 0 ; 
and that of the solid relatively to the liquid, per unit of velocity and density, is represented 
by — D, D denoting the displacement of the solid (that is, the volume of liquid which 
it displaces, and also the mass of the solid supposed equal to that of the displaced liquid). 
Thirdly, let the common centre of mass of the liquid and solid be taken as a fixed 
point, and let the momenta of the liquid and solid relatively to that point be taken. 
Those momenta are equal and opposite — that of the liquid being positive, and that of 
2 r 2 
