Sia William Rowan Hamilton on Fluctuating Functions. 317 



Lagrange appears to have been the first who was led (in connexion with the 

 celebrated problem of vibrating cords) to assign, as the result of a species of in- 

 terpolation, an expression for an arbitrary function, continuous or discontinuous 

 in form, between any finite limits, by a series of sines of multiples, in which the 

 coefficients are definite integrals. Analogous expressions, for a particular class of 

 rational and integral functions, were derived by Daniel Bernouilli, through 

 successive integrations, from the results of certain trigonometric summations, 

 which he had characterized in a former memoir as being incongruously true. No 

 farther step of importance towai'ds the improvement of this theory seems to have 

 been made, till Fourier, in his researches on Heat, was led to the discovery of 

 his well known theorem, by which any arbitrary function of any real variable is 

 expressed, between finite or infinite limits, by a double definite integral. Poisson 

 and Cauchy have treated the same subject since, and enriched it with new views 

 and applications ; and through the labours of these and, perhaps, of other writers, 

 the theory of the development or transformation of arbitrary functions, through 

 functions of determined forms, has become one of the most important and inte- 

 resting departments of modern algebra. 



It must, however, be owned that some obscurity seems still to hang over the 

 subject, and that a farther examination of its principles may not be useless or un- 

 necessary. The very existence of such transformations as in this theory are 

 sought for and obtained, appears at first sight paradoxical ; it is difficult at first 

 to conceive the possibility of expressing a perfectly arbitrary function through any 

 series of sines or cosines ; the variable being thus made the subject of known and 

 determined operations, whereas it had offered itself originally as the subject of 

 operations unknown and undetermined. And even after this first feeling of pa- 

 radox is removed, or relieved, by the consideration that the number of the opera- 

 tions of known form is infinite, and that the operation of arbitrary form reappears 

 in another part of the expression, as performed on an auxiliary variable ; it still 

 requires attentive consideration to see clearly how it is possible that none of the 

 values of this new variable should have any influence on the final result, except 

 those which are extremely nearly equal to the variable originally proposed. This 

 latter difficulty has not, perhaps, been removed to the complete satisfaction of those 

 who desire to examine the question with all the diligence its importance deserves, 

 by any of the published works upon the subject. A conviction, doubtless, may 



