88 Prof. Challis on the Principles of Hydrodynamics. 



the numbers 1, 2, 8, &c. to ^ or — ^ — , according as n is even 



or odd ; and the last terra gives every pair following the (w + l)th 

 by substituting for /?, 1, 2, 3, &c. ad infinitum. Let /i be a very 

 large number, and suppose that a^=n^. Then we have 



+ 1«. 2». 3' . . . n«/= ^^ ^ "^^-^ 



(■-.-)-('-!)'-('-*^) 



.■■.((i^.a)--i) 



If/) be taken so that 2j9— 1 is comparable with w, the factors 



(-i)'(-D'-(-*?iy 



and 



(■*i)-('*S--('-*?^) 



become very small, and the terms which they multiply become 

 inconsiderable with respect to those for which 2/? — 1 is not com- 

 parable with n. Now for the latter terms the right-hand side of 

 the above equality is satisfied, if quantities inferior by two degrees 

 at- least to the terms of the highest order be omitted, as will 

 appear by expanding the factors 



and 





each to two terms. But we have seen above that if a^=w^, the 

 nth and {w+l)th terms are the greatest. It hence appears 

 that the value n^ of a^ causes the principal terms to vanish 

 approximately. And as these terms cannot be made to vanish 

 approximately by terms that are inconsiderable with respect to 

 them, it follows that the value w of a approximates more and 

 more to a root of the equation /=0 in proportion as n is larger. 

 By parity of reasoning w H- 1, » + 2, &c. are roots of the same 

 equation, n being indefinitely large. We may therefore conclude 

 that the infinite roots of the equation /=0 form an arithmetic 

 series of which the common diftcrcnce is unity. 

 By precisely analogous reasoning I have found that the value 



