318 Sir William Rowan Hamilton on Fluctuating Functions. 



be attained, that the results are true, but something Is, perhaps, felt to be still 

 wanting for the full rigour of mathematical demonstration. Such has, at least, 

 been the impression left on the mind of the present writer, after an attentive 

 study of the reasonings usually employed, respecting the tranformations of arbi- 

 trary functions. 



PoissoN, for example, in treating this subject, sets out, most commonly, with 

 a series of cosines of multiple arcs ; and because the sum is generally indetermi- 

 nate, when continued to infinity, he alters the series by multiplying each term by 

 the corresponding power of an auxiliary quantity which he assumes to be less 

 than unity, in order that Its powers may diminish, and at last vanish ; but, in 

 order that the new series may tend Indefinitely to coincide with the old one, he 

 conceives, after effecting Its summation, that the auxiliary quantity tends to be- 

 come unity. The limit thus obtained is generally zero, but becomes on the con- 

 trary Infinite when the arc and Its multiples vanish ; from which It Is Inferred by 

 PoissoN, that if this arc be the difference of two variables, an original and an 

 auxiliary, and if the series be multiplied by any arbitrary function of the latter 

 variable, and integrated with respect thereto, the effect of all the values of that 

 variable will disappear from the result, except the effect of those which are ex- 

 tremely nearly equal to the variable originally proposed. 



PoissoN has made, with consummate skill, a great number of applications of 

 this method ; yet It appears to present, on close consideration, some difficulties 

 of the kind above alluded to. In fact, the introduction of the system of factors, 

 which tend to vanish before the Integration, as their Indices increase, but tend to 

 unity, after the integration, for all finite values of those indices, seems somewhat 

 to change the nature of the question, by the Introduction of a foreign element. 

 Nor is it perhaps manifest that the original series, of which the sum is indeter- 

 minate, may be replaced by the convergent series with determined sum, which 

 results from multiplying Its terms by the powers of a factor Infinitely little less 

 than unity ; while it is held that to multiply by the powers of a factor Infinitely 

 little greater than unity would give an useless or even false result. Besides there is 

 something unsatisfactory In employing an apparently arbitrary contrivance for 

 annulling the effect of those terms of the proposed series which are situated at a 

 great distance from the origin, but which do not themselves originally tend to 

 vanish as they become more distant therefrom. Nor is this difficulty entirely 



