366 Sir W. R. Hamilton on the Numbers of Bernoulli. 



(28.) for A 2 and B 2 „__ j» by the process of the last article, 



(on the ground that the parts omitted or introduced thereby 



must at least nearly destroy each other, through what may be 



called the " principle of fluctuation,") are now seen to have 



produced no ultimate error at all, their mutual compensation 



being perfect; a result which may tend to give increased 



confidence in applying a similar process of approximation, or 



transformation, to the treatment of other similar integrals; 



although the logic of this process may deserve to be more 



closely scrutinized. Some assistance towards such a scrutiny 



may be derived from the essay on (i Fluctuating Functions," 



which has been published by the present writer in the second 



part of the nineteenth volume of the Transactions of the Royal 



Irish Academy. 



6. It may be worth while to notice, in conclusion, that the 



property marked (7.) of the definite integral (1.), enables us to 



. cos^px — x) ^ . . . . . A . 



change — to sin 2p x tan x, in the equations(14.), 



(15.), (19.)> (28.); so that the pth Bernoullian number may 

 rigorously be expressed as follows : — 



n , (-l) p - 1 .1.2...3p 



»2 P -l- 2 2 ^- 1 (2 2 ^-l)* 



/»" . (s\nx\ 2p . n 



I dx\ J sin 2 p x tan x ; 



under which form the preceding deduction of its transfor- 

 mation (29.) admits of being slightly simplified. The same 

 modification of the foregoing expressions conducts easily to 

 the equation 



(30.) 



log « = 



e + e 



vfo d * 



tan x tan ' 



h sin x sin 2 x 



x — h sin x cos 2 x 



in which tan is a characteristic equivalent to arc tang., 

 and which may be made an expression for log sec x, by 



merely changing the sign of h in the last denominator; and 

 from this equation (31.) it would be easy to return to an ex- 



h 

 e — e 



pression for the coefficients in the development of — - — 



e + e 

 or in that of tan h, and therefore to the numbers of Ber- 

 noulli. Those numbers might thus be deduced from the 



