92 Letter from Mr. Ivory tu Professor Airy. 



of — and perform the operations indicated, it will appear that 

 this equation is exact, viz. 



In the particular case when a = 1 and p = 1, the last factor 

 but one on the right side is infinitely great on the supposition 

 that d7H has a constant thickness; and, although the last fac- 

 tor is evanescent for all values of p less than 1, no conclusion 

 can be drawn respecting the value of the integral without a 

 particular examination of the case. This sufficiently accounts 

 for the difficulty of Lagrange. It follows too that the integral 

 will be evanescent and the equation exact, when the thickness 

 of d 7)1 is divisible by 1 —p, or by the square of the evanescent 

 distance from the attracted point : for the factor, which was be- 

 fore infinite, will now have a finite value. In reality this is the 

 limitation laid down in the demonstration, Mec.Celeste, liv. 1 1""% 

 No. 2, which was first published in 1816. It is remarkable 

 that the author did not perceive, or at least does not notice, 

 that the equation, which was first supposed to be unbounded 

 in its application, is confined, by the new demonstration, 

 within very narrow limits. But we shall attain the utmost 

 generality of which the equation is capable, if we write 

 {y'~ y) (^^ for din, y' — y being the thickness of the molecule: 

 then, 



"2' '^ "■ da. —J A/'2a(l-?') ' 7 * / " / ' 



and the equation is true when y^ — y\% divisible by v' 1 — p, 

 and in no other case whatsoever. 



It will not be necessary to notice particularly the investiga- 

 tion which Lagrange has given of the differential equation : 

 because it depends upon a transformation, and an integration 

 by parts, which cannot be executed unless j/' be a rational 

 function of three rectangular coordinates. 



May we now venture to conclude, that this analytical me- 

 thod, although it has passed through the hands of Laplace, La- 

 grange and Poisson, has never been well undei'stood ; that it 

 has been founded on inconclusive demonstrations; and, when 

 its principles are clearly unfolded, that it falls entirely within 

 the ordinary rules of investigation ? 



I have the honour to be, &c. &,c. 



July 9, 1827. James Ivory. 



XVIII. Letter 



