Mr. Ivory's Remarks on M. Poisson's Memoir. 329 



take place unless two conditions, or laws, which I need not 

 here repeat, are both fulfilled, ot which one only is necessary 

 according to the usual doctrine*. 



There is a remarkable proof of the deficiency of the usual 

 theory in M. Poisson's Memoir. He apphes his analysis to a 

 homocreneous fluici mass, revolving upon an axis, and nearly 

 spherical +. When the square of the centrifugal force is neg- 

 lected, he finds that the figure of the fluid must be an ellipti- 

 cal spheroid, agreeing with the solutions of Legendre and 

 Laplace. But, on attempting to carry the approximation 

 further, the method tails; all that can be known is, that there 

 is only one figure which will satisfy the equations :-M«25 ce 

 vrocede nc sauroit (Uterminer davantage ce solidei. Now, wliat 

 is the reason of this ? It cannot be the want of mathematical 

 methods; for the symbols are all arranged, and ready to obey 

 the directions of the analyst. The truth is, there is no prin- 

 ciple to cTovern the calculation after the first step, ihe ma- 

 chinery ?s sufficient, and ready prepared ; but it cannot be set 

 to woi-k, because there is no fulcrum for its support. In or- 

 der to supply the defects of his method, M. Poisson has re- 

 course to the elliptical spheroid, which is known to satisfy the 

 conditions of the problem; and he infers that his series for 

 the radius of the solid, must coincide with the expansion ot 

 the radius of the ellipsoid §. Now it is far from clear that he 

 is ricrht in this inference. If I take in both my conditions, 

 and'thence deduce the resulting figure of equilibrium, there 

 is no doubt that the radius, to whatever length the expansion 

 is carried, will coincide with an elliptical spheroid; because 

 this is the only figure deducible from the premises. But, it 

 I leave out one of my two conditions, and attempt to solve the 

 same problem by means of the other alone, which is exactly 

 what M. Poisson has done, it is next to certam that the new 

 computation will not agree with the former one. 



There are rio direct objections to my theory; but it stands 



» In the Phil. Trans. 1826, p. 557, there is a note of Mr. Airy, very in- 

 jurious to me. He is treating of ^pheroids of variable density, and evi- 

 dcntly misapprehends my conditions of equ.l.bnnm, ^vh,ch I have a va s 

 lin.ited to the case of homogeneity. The R. S. are not •"esponsible tor e 

 accuracy of what they publish : but I apprehend few instances w.l be f md 

 so injurious to an individual, cast upon the pubhc on the ^"tl'^'-'^y °J, "^ J 

 assertion, and arising from mistaken notions. But I console myself bcc.use 

 1 know with the certainty of demonstration, that Mr. Airy s problem ad- 

 mitting that any practical utility could be attached to it, is "» /«';"'' «"^ 

 diat it^annot possibly be solved except by my theory =|.- ;"''>(" J^'/j? 

 the hel,. of that law with which he so flippantly finds taut. ^ hat a dit- 

 ference between the supercilious importance of the Cambridge liotes.or, 

 and the candid exiiositions of M. Poisson ! r ,1 • i 171; 



t Conn, dcs rL 1829. p. 371. X Ibid. p. 373. $ Ibul. p. 375 

 New Series. Vol. 1. No. 5. Mai/ 1827. 2 U opposed 



