237 



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 dif- 

 ference of values of these positive and negative portions, cor- 

 responding to a 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 it 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 foreseen 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 pre- 

 sent 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 appHcable 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 



* Sir William Hamilton's Essay on Fluctuating Functions, will be found in 

 the Second Part of volume xix. of the Transactions of the Academy. 



