330 Mr. Ivory's Remarks on M. Poisson's Memoir. 



opposed to the splendid analytical processes that have been 

 so long and so unsparingly admired. According to my vievt^, 

 there can possibly be but one figure of equilibrium of a homo- 

 geneous planet in a fluid state; and in fact, this compre- 

 hends all that geometers have been able to accomplish in this 

 question. The usual theoi'y advances one step in one parti- 

 cular case; and then it leaves the geometer in the lurch, with- 

 out his being able to explain the reason of the failure. Be- 

 yond this it has been entirely inefficient : — Quand la masse 

 Jluide n'cst pas assujettie a differer tres peu de la sphere^ les 

 geometres liont point encore determine. Vespece de Jigure qui 

 satisjait a V equation d'eqiiilibre *. In an elliptical spheroid 

 ill equilibrio, it is known that the rotatory velocity is limited, 

 being contained between zero and a maximum quantity ; so 

 that there are two different figures that have the same rota- 

 tion. On this ground M. Poisson makes an objection, which 

 I notice the more willingly, because it does not turn upon any 

 technical point of analysis. Si Vellipsoide etait la seule figure 

 qui eut cette propriete, il en resulterait cette consequence singu- 

 liere, que Vequilibre serait impossible pour une rapidite de la 

 rotation, qui riest cependant pas celle ou le Jluide commencerait 

 a sc dissiper*. 



Suppose a homogeneous mass of fluid, at rest, in equilibrio, 

 and consequently spherical in its figure: conceive a great 

 circle of the sphere extending indefinitely, and an axis, or 

 diameter, perpendicular to the great circle. Now let a velo- 

 city of rotation about the axis be communicated to the fluid 

 sphere : I impose no restriction to the degree of the velocity, 

 except that it must not be such as to dissipate the particles, 

 which must retain their continuity. The rotatory motion will 

 cause the fluid to recede from the axis, and to subside at the 

 poles ; and to these effects there would be no limit, if the cen- 

 trifugal force were not opposed by that part of the attraction 

 of the particles which is directed perpendicularly to the axis. 

 At a certain degree of oblateness the two opposite forces will 

 be equal ; and although the recession of the fluid from the 

 axis will not immediately cease at this point, yet it will soon 

 be entirely arrested. The figure of the fluid will now return 

 in an opposite direction, becoming less oblate, and passing a 

 little beyond the limit at which the two forces are equal. The 

 fluid will thus oscillate about a state of equilibrium ; and if we 

 admit any tenacity or friction of the particles, the oscillations 

 will gradually decrease, and finally settle in a permanent figure. 

 But it is to be principally observed that, whenever the fluid 



• Conn, des Term 1829, p. 375. 



recedes 



