"WITHDRAWN FROM THE ACTION OP GRAVITY. 653 



first dilatation ; a third at the origin of the second constriction ; 

 a fourth at the origin of the second dilatation, and so on ; so that 

 all the sections of the even series will occupy the origins of the 

 dilatations, all those of the odd series the origins of the constric- 

 tions. The interval comprised between two consecutive sections 

 of the odd series will therefore include a constriction and a dilata- 

 tion ; and as the figure begins with a constriction and terminates 

 with a dilatation, it is clear that its entire length will be divided 

 into a whole number of similar intervals. In consequence of 

 the exact regularity which we have supposed to exist in the 

 transformation, all the intervals in question will be equal in 

 length ; and as the moment at which we enter upon the con- 

 sideration of the figure may be taken arbitrarily from the origin 

 of the phaenomenon to the maximum of the depth of the con- 

 strictions, it follows that the equality of length of the intervals 

 subsists during the whole of this period, and that, consequently, 

 the sections which terminate these intervals preserve during this 

 period perfectly fixed positions. Besides the parts of the figure 

 respectively contained in each of the intei'vals undergoing iden- 

 tically and simultaneously the same modifications, the volumes 

 of all these parts remain equal to each other ; and as their sum 

 is always equal to the total volume of the liquid, it follows that, 

 from the origin of the transformation to the maximum of depth 

 of the constrictions, each of these partial volumes remains inva- 

 riable, or in other words, no portion of liquid passes from any 

 one interval into the adjacent ones. Thus, at the instant at 

 which we consider the figure, on the one hand, the two sections 

 which terminate any one interval will have preserved their primi- 

 tive positions and their diameters ; and on the other hand, these 

 sections will not have been traversed by any portion of liquid. 

 Matters will then have occurred in each interval in the same 

 manner as if the two sections by which it is terminated had been 

 solid discs. But the transformation cannot ensue between two 

 solid discs, if the proportion of the distance which separates the 

 discs to the diameter of the cylinder is less than the limit of 

 stability ; the proportion of the length of our intervals and the 

 diameter of the cylinder cannot then be less than this limit. Now, 

 the length of an interval is evidently equal to that of a division ; 

 for the first, in accordance with what we have seen above, is the 

 sum of the lengths of a dilatation and a constriction ; and the 

 second is the sum of the lengths of a dilatation and two semi- 



VOL. V. PART XXI. 2 Y 



