340 On the Expansion of Elastic Fluids. 
one fourth the subtangent of the fluid, the absolute velocities of 
the several grades respectively will be expressed by the respective 
terms of the series v, 2v, 3v, 4v, 5v, €c. 
Since one element =s by measure*passes from each grade into 
the next, and becomes = 2s in the next, the length of each grade 
at the end of the first instant =2s—s=s. That is, the length 
of each grade is equal to that of the original element; and the 
place of the first grade is that which was occupied by the original 
element ; the other grades succeeding it in continuous order. 
Having now ascertained the state of things at the end of the 
first instant, let us inquire what takes place in the second instant. 
It is obvious that during the second instant the front of the 
second element of the column, and also the front of each grade 
respectively is a point of expansion from which one element =s 
by measure passes into the next grade. ‘Thus in the second 
instant each grade receives an addition of 2s to its rear and loses 
s from its front. The same takes place in every succeeding 
instant. Since the increment of the length of the grades for each 
‘instant is s, the velocity of the increase is v. The length of the 
grades is therefore always equal to the space through which the 
point of expansion has receded in the column. Thus while the 
length of the grades increases with the uniform velocity v, their 
number, velocity and density remain unchanged. Consequently 
no other gradations of density can exist in front of a column ex- 
panding into a vacuum, but those which correspond to the terms 
of the infinite geometrical series - 1°38’ 16 &3 and no other 
gradations of velocity but those which correspond to the terms of 
the infinite arithmetical series v, 2v, 3v, 4v, &c. 
_ The point of expansion in the column recedes with the velo- 
city v; and since the length of the first grade is always equal to 
the space throngh which that point of expansion has moved, it 
follows that the point of expansion from the first into the second 
grade is stationary. And since the second grade increases 12 
length with the velocity v, the third point of expansion moves 
forward with the velocity v; and since all the other g 
increase in length with the same velocity v, the velocities of the 
several points of expansion will be expressed by the following 
series —v, 0, v, 2v, 3v, 4v, &e. 
In order to give a synopsis of the results to which we have 
come, let AB be acolumn of fluid of the density D, expanding 
into a vacunm toward C. Let the velocity due to a height equal 
to one fourth of the subtangent of the fluid be v. Suppose ex- 
pansion to have commenced at B, and the point of expansion to 
have receded to any distance m. Set off from B an infinite num- 
ser of spaces Ba, ab, bc, ed, &c., each equal to Bn. Then the 
points 7, B, a, b, c, d, &c., are the places of the points of expan- 
