Letter from Mr. Ivory to Pi-ofessor Airy. 91 



that to this class the method likewise extends. But this hap- 

 pens, not because the analysis is more extensive in its prin- 

 ciples than I have represented it to be, but because whatever 

 is genei'ally true of an indefinite number of approximations 

 tending to a limit, must be true of that limit. 



The purpose of this letter is now accomplished by what I 

 have written, which fully unfolds the principle and the extent 

 of the analytical method of Laplace ; but, before concluding, 

 it may not be improper to add a few words concerning the 

 demonstration which the author of the M.ec. Celeste has given 

 of his fundamental equation. 



The demonstration in the Mec. Celeste, liv. 3™*^, No. 10, 

 rests upon this formula, 



1 



1 1 "-7 



which, it is assumed, is ahsoays equal to zero when « = 1. Now 



if we put V = f—j-, all the molecules being near the surface 



of the sphere, and observe that the differential equation is true 

 of every individual molecule, we shall have, 



2 da. 



This demonstration is independent of the nature of the mole- 

 cule. It is true that Laplace makes the sphere touch the 

 spheroid at the atti'acted point ; but no stress is laid upon this 

 construction in the process of finding the equation : and it 

 seems only intended by it to show that the gravitation per- 

 pendicular to the two surfaces at the point of contact, may be 

 reckoned equal to the gravitation tending to the centre of gra- 

 vity of the spheroid. 



It is certain at least that Lagrange understood the demon- 

 stration as I have explained it. For he supposes that all the 

 molecules have their thickness equal to a small constant quan- 

 tity, and he finds that the equation is not exact. He dis- 

 cusses this point at some length, with his usual clearness and 

 precision*; he calls it, ime difficulte singuliere ; and he ex- 

 presses some admiration, as well he might, that a number of 

 zeros should produce an aggregate not equal to nothing. 



But there is a defect in the demonstration of Laplace, which 

 is also the origin of the difficulty of Lagrange; namely, their 

 supposing that the formula (B) is always exact. It is true for 

 every value of ]) less than 1 ; but when y^ = 1, instead of be- 

 ing evanescent, it is infinitely great. If we substitute the value 



' Joiinial dc C Ecolc I'olijl. IT)"" cahicr, p. hT. 



N 2 of 



