[ MACGREGOR | HYPOTHESES OF ABSTRACT DYNAMICS 91 
an additional hypothesis. ‘Thus (to cite a few works which happen to be 
at hand) Rankine! expressly formulates the principle of continuity as an 
axiom; Kelvin and Tait, Minchin Lord Rayleigh,* and Greenhill® 
deduce the equation from the impossibility of the annihilation and the 
creation or generation of matter ; Duhamel,’ Besant,’ and Bassett * obtain 
it from the continuity of the motion of a fluid; while Poisson,’ Schell,” 
Kirchhoff," Besant,” Lamb," who regards it as “a kinematical relation 
obtained by expressing the constancy of mass,’ and Tait,’ who regards 
it as “the mathematical expression of the fact that the number of points” 
[of a deformable figure] “is not altered’” by deformation, all obtain it 
more or less clearly from the constancy of mass, which is asserted or 
assumed to be evident or obvious. 
That the equation of continuity involves no assumption in addition 
to those introduced above, however, may readily be shown. Why it is 
needed in the solution of many problems becomes apparent when we note 
that in obtaining the equations of motion the Second Law of Motion is 
only partially applied. That law tells us that the quotient of the force 
acting on a body by the acceleration it produces is constant, the value of 
the constant being called the mass of the body. Now, in obtaining the 
equations of motion given above,the sum of the components in the x axis of 
ou 
the forces acting on the element dx dy dz is put equal to Psp dx dy dx, 
p dx dy dz being the mass of the element; and similarly for the y and z 
axes. There is thus nothing in the equations to express the constancy of 
the quotient of force by acceleration ; the Second Law of Motion is thus 
only partially applied, and for a complete application of it an additional 
equation is necessary, viz., the statement in some form or other that the 
p dx dy dz of the equations of motion is constant. Thus the equation of 
continuity involves no hypothesis in addition to those mentioned above. 
It is simply a partial application of Newton’s Second Law, which, 
together with the partial application of the same law made in obtaining 
the equations of motion, constitute a complete application of the law. 


1 ** Applied Mechanics,” 9th ed., p. 411. 
2“ Treatise on Nat. Phil.,” vol. i., part 1, § 191. 
*“ Uni-planar Kinematics of Solids and Fluids,” 1882, p. 142. 
4 “Theory of Sound,” 1878, vol. ii., p. 3. 
“Encyclopedia Britannica,” 9th ed., art. Hydromechanics, p. 445. 
5 ** Cours de Mécanique,-’ 3me ed., 1863, tome ii., p. 284. 
7 ** Hydro-mechanies,” 8rd ed., 1877, p. 168. 
8“ Hydro-mechanies,” vol. i., 1888, p. 7. 
“Traité de Mécanique,” 2me ed., 1833, tome ii., p. 669. 
0“ Theorie d. Bewegung u. d. Kriifte,” 1870, p. 939, 
 * Mechanik,” 2te Aufl., 1877, p. 129. 
2 * Hydro-mechanics,” 3rd ed., 1877, p. 170. 
15 “* Mathematical Theory of the Motion of Fluids,” 1879, p. 5. 
4“ Encyclopedia Britannica,” 9th ed., art. Mechanics, § 94. 
ot 
