320 SiK William Rowan Hamilton on Fluctuating Functions. 



PoissoN has given on the subject of these transformations. For example, 

 although Fourier, in the proof just referred to, of the theorem called by his 

 name, shows clearly that in integrating the product of an arbitrary but finite 

 function, and the sine or cosine of an infinite multiple, each successive positive 

 portion of the integral is destroyed by the negative portion which follows it, if 

 infinitely small quantities be neglected, yet he omits to show that the infinitely 

 small outstanding difference of values of these positive and negative portions, 

 corresponding to the single period of the trigonometric function introduced, is 

 of the second order; and, therefore, a doubt may arise whether the infinite 

 number of such infinitely small periods, contained in any finite interval, may not 

 produce, by their accumulation, a finite result. It is also desirable to be able to 

 state the argument in the language of limits, rather than in that of infinitesimals ; 

 and to exhibit, by appropriate definitions and notations, what was evidently fore- 

 seen by Fourier, that the result depends rather on the fluctuating than on the 

 trigonometric character of the auxiliary function employed. 



The same view of the question had occurred to the present writer, before he 

 was aware that indications of it were to be found among the published works of 

 Fourier ; and he still conceives that the details of the demonstration to which 

 he was thus led may be not devoid of interest and utility, as tending to give 

 greater rigour and clearness to the proof and the conception of a widely applicable 

 and highly remarkable theorem. 



Yet, if he did not suppose that the present paper contains something more 

 than a mere expansion or improvement of a known proof of a known result, the 

 Author would scarcely have ventured to offer it to the Transactions* of the 

 Royal Irish Academy. It aims not merely to give a more perfectly satisfactory 

 demonstration of Fourier's celebrated theorem than any which the writer has 

 elsewhere seen, but also to present that theorem, and many others analogous 

 thereto, under a greatly generalized form, deduced from the principle of fluctu- 



* The Author is desirous to acknowledge, that since the time of his first communicating the pre- 

 sent paper to the Royal Irish Academy, in June, 1840, he has had an opportunity of entirely re- 

 writing it, and that the last sheet is only now passing through the press, in June, 1842. Yet it may 

 be proper to mention also that the theorems (A) (B) (C), which sufficiently express the character of 

 the communication, were printed (with some slight differences of notation) in the year 1840, as part 

 of the Proceedings of the Academy for the date prefixed to this paper. 



