jMr. Ivory's Reply to Professor Airy. 1 7 



terized by the terms I have used. I should be sorry to be 

 found a great trespasser on this occasion ; but, for many years, 

 I have been very frequently attacked without rhyme or reason, 

 and with no regard to scruples: and this will, I hope, be 

 deemed a sufficient apology for any warmth of language, more 

 especially to a gentleman who, it appears from his paper, knows 

 full well to do himself justice. 



What I read in the last Phil. Mag., p. 447, induced me to 

 examine my investigation in the preceding Number, pp. 327, 

 S28 ; and I find that I have drawn a wrong inference from 



h k 



my analysis. The evanescent fractions — , — (/z, k, g all va- 



s s 



nish together withy) are in every case equal to zero ; so that 

 my reasoning is entirely favourable to M. Poisson's proposition. 

 As my investigation is free from precarious assumptions, 1 will 

 briefly state the steps of it, referring to the place cited for an 

 explanation of the symbols. We have, 

 y = ^ + A7? + B^; 

 consequently, 



1-rJp 1-x J p "^ 2vJ /3 



Now, <?s = sin 9'(f fl'rfvj/' = %v!\^dhdh\ and, the value of^s 

 being substituted, the second term on the right hand integrated 

 as at the place cited, is always equal to zero ; so that we have 



_J_ r gy'ds _ _y_ / r gds _ 



It J /3 2^y /3 - y- 



Admitting therefore that the proposition is proved in the most 

 general sense, let us examine the consequence. 



The development is likewise proved in the most general 

 sense. For there is no objection to the process by which the 

 following formula is deduced from the general theorem, viz. 



^aiza^y = \- —^ ; h &c. 



Now this is the development. For we have, 



and as Q^'' is a function of /x, \/ I — [i-. cos co, V I — 1».\ sin a>, 



it is plain that U^*^ will be a like function, the coefficients alone 

 being changed by the integrations. The development may 

 always be found, namely, by performing the re(juisite integra- 

 tions for every term. We must admit too that it is unicjue ; 

 lor every term is a definite integral, involving nothing pre- 

 New Series. Vol. 2. No. 7. July 1827. D carious 



