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XII.— Ow Fluctuating Functions. By Sir William Rowan Hamilton, 

 LL. D., P. R. I. A., F. R, A. S., Fellow of the American Society of Arts 

 and Sciences, and of the Royal Northern Society of Antiquaries at Copen- 

 hagen ; Honorary or Corresponding Member of the Royal Societies of 

 Edinburgh and Dublin, of the Academies of St. Petersburgh, Berlin, and 

 Turin, and of other Scientific Societies at hom^, and abroad ; Andrews' 

 Professor of Astronomy in the University of Dublin, and Royal Astronomer 

 of Ireland. 



Eead June 22nd, 1840. 



The paper now submitted to the Royal Irish Academy is designed chiefly to 

 invite attention to some consequences of a very fertile principle, of which indica- 

 tions may be found in Fourier's Theory of Heat, but which appears to have 

 hitherto attracted little notice, and in particular seems to have been overlooked 

 by PoissoN. This principle, which may be called the Principle of Fluctuation, 

 asserts (when put under its simplest form) the evanescence of the integral, taken 

 between any finite limits, of the product formed by multiplying together any two 

 finite functions, of which one, like the sine or cosine of an infinite multiple of an 

 arc, changes sign infinitely often within a finite extent of the variable on which it 

 depends, and has for its mean value zero ; from which it follows, that if the other 

 function, instead of being always finite, becomes infinite for some particular values 

 of its variable, the integral of the product is to be found by attending only to the 

 immediate neighbourhood of those particular values. The writer is of opinion 

 that it is only requisite to develope the foregoing principle, in order to give a 

 new clearness, and even a new extension, to the existing theory of the transfor- 

 mations of arbitrary functions through functions of determined forms. Such is, 

 at least, the object aimed at in the following pages ; to which will be found 

 appended a few general observations on this interesting part of our knowledge. 



