French National Injlitute. 299 



As this hypothec's gives a refult too great, C. Biot then 

 refumes the queftion in an inverfe order, and, from the ob- 

 ferved velocity of found, endeavours to find what ought to 

 be the quantity of caloric abandoned by the air, when it is 

 reduced by compretfion to the half of its volume; and he 

 finds that the fame quantity would raife Reaumur's thermo- 

 meter to about 69 . 



C. Biot has communicated alfo to the craft refearches on 

 the attraction of fpheroids. This fubject, treated firlt in a fyh- 

 thetic manner bv Maclaurin, was a long time the quickfand 

 of analyfis, which, however, in the hands of Lagrange, Le- 

 gehdre, and Laplace, has fucceffively affumed afuperiority over 

 lynthcfis, and conduced torefnlts which could not have been 

 obtained without its athftance. But there remained in the 

 analytical demonttrations of the principal theorems on this 

 fubject, a complication which C. Biot made to difappear 

 in a very happy manner, by combining a theorem, for which 

 we are indebted to Lagrange, with a partial differential equa- 

 tion, found by Laplace, and applying to this equation a pro- 

 cefc, which he prefented himfelf fome years ago to the clafs, 

 to integrate by feries partial differential equations. 



The equation to which we allude is among three of the diffe- 

 rential co-efficients of the fecond order of the function which 

 expieffes the fum of the molecular of the fpheroid divided by 

 their diftance from the point attracted ; its integration gives 

 for this quantity a feries containing two arbitrary functions 

 arranged according to the powers ot one of the co-ordinates 

 of the point attracted. By fucceffively taking, in regard to 

 each of thefe variable quantities, the differential co-efficients 

 of the feries which expreiTes the attractions exercifed by the 

 fpheroid in a direction parallel to the axes of the co ordi- 

 nates, C. Biot obtained developments of thefe attractions, de- 

 termined entirely by three quantities, independent of the va- 

 riable one, according to the powers of which the develop- 

 ments are arranged. 



It thence remits, iff, that to have the attractions of any 

 fpheroid on any point of fpace, it is fufficient to take at pleafure 

 a plane, and to calculate the attractions of the fpheroid on the 

 points (hinted in that plane: thofe which are within the 

 fpheroid will determine the general expreffion of its attraction 

 on the interior points; the other will determine thofe which 

 belong to the exterior points. 



2d, That if two fpheroids are fuch that their attractions 

 on all the points of the fame plane parallel to three rectan- 

 gular axes «*re to each other in a conftant ratio, the attrac- 



U 4 tions 



