158 Compressible and Non-Homogeneous Masses [CH. vn 



With a view to discovering dynamical points of bifurcation on the sequence 

 of configurations determined by equations (411), let us compare this motion 

 with a slightly varied motion in which the particle which in the original 

 motion was at #, y, z at time t is, in the varied motion, at the point 



a? + f , y + *7, z + % 

 at time t. 



In the varied motion the particle just specified will have components of 

 acceleration 



'"V* f" 



so that the particle which is at x, y, z at time t in the varied motion will 

 have components of acceleration 



Let the values of X and p at the point x, y, z be changed, in the varied 

 motion, to X + $X and p + Sp. Then the equations by which the varied 

 motion is governed will be 



On subtracting corresponding sides of equations (411) and (412) we obtain 



......... (413), 



which is an equation of motion for all small displacements which can be super- 

 posed on to the original motion while still conforming to the laws of dynamics. 

 The original motion must be determined from the three equations (411). 

 Equation (413) and its two companions will then determine the dynamical 

 points of bifurcation on this motion. 



Equations (411) cannot be solved in detail, so that an exact knowledge of 

 the dynamical points of bifurcation determined by equation (413) cannot be 

 obtained. But a knowledge of the general nature of the solution of equation 

 (413) can be obtained from a consideration of the simple case in which f x , f y 

 and f z are all constants, so that the jet is supposed to move with uniform 

 acceleration e.g. as though moving under a uniform gravitational field. 



Equation (413) now reduces to 



and there are two similar equations. Differentiating with respect to x, y, z 

 and adding, we obtain 



