Professor Airy in reply to Mi-. Ivory. 447 



monstrated, and for all kinds of spheroids differing little from 

 a sphere. The only limitation of its generality is, that the sine 

 or tangent of the angle made by the spherical and spheroidal 

 sm-faces at their intersection, must be expressed by a finite 

 multiple of u ; which condition is satisfied when y is expressed 

 by any function, rational or irrational, that never makes 



^y ^„ * ^y 



-v/ 1 — u,- . -r- or — = . -~- infinite. I have only to add, 



that this part of the paper is little more than a filling-up of 

 the sketch given by Laplace in one of the last books of the 

 Mt'canique Celeste. 



I cannot at present enter on the discussion of a very nice 

 and abstruse point : I shall merely remark, that the difficulties 

 which Mr. Ivory has found (see his paper, Phil. Trans. 1812, 

 p. 16), appear to arise from the separation of 3/ — j/ into two 

 parts. For the rest I must beg leave to refer the reader to my 

 paper in the Cambridge Transactions, vol. ii. " Now," says 

 Mr. Ivory, " admitting that the equation in question is accu- 

 rately and numerically proved, it seems impossible to deny 

 that the series of terms deduced from it is numerically equal to 

 the distance between the surfaces of the sphere and spheroid." 

 With this I perfectly agree : but Mr. Ivory afterwards says, 

 " Mr. Professor Airy, by supporting the fundamental equation 

 without restricting it, and at the same time denying the un- 

 avoidable consequence, has only introduced new inconsist- 

 encies," &c. I can only infer from this that Mr. Ivory has 

 not read the whole of my paper. However little the trouble 

 of reading it might be repaid, it is not right to make such 

 remarks on the connection of the first and the last parts, with- 

 out examining or alluding to the subject which occupies the 

 body of the paper. In the beginning I have endeavoured to 



show that the equation — a — — = — — + — is generally 

 true. From this the equation ^u.iza'^y = 1 '—^ — + 



— ^-^ — + &c. is derived by an unobjectionable process. But 



this equation as it stands is useless, unless we can resolve 

 4 a TT a'^y into a series of terms, distinguished by the same pe- 

 culiarities which separate those on the other side of the equa- 

 tion. If it is not possible to resolve Anxna-y into more than 

 one such series, the corresponding terms must be e(]ual : if it 

 is possible to do it in more than one way, nothing can be in- 

 ferred from the equation, but the equality of the whole quan- 

 tity on one side to the whole (juantity on tiic otiier side. It 

 is tlicrefore necessary to prove that tliis resolution can be if- 



li'CtC'll 



