233 



The following general observations are extracted, on the 

 nature and history of this branch of analysis:- — 



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 interpolation, an expres- 

 sion for an arbitrary function, continuous or discontinuous in 

 form, between any finite limits, by a series of sines of multi- 

 ples, in which the coefficients are definite integrals. Analo- 

 gous expressions, for a particular class of rational and inte- 

 gral functions, were derived by Daniel Bernouilli, through 

 successive integrations, from the results of certain trigono- 

 metric summations, which he had characterized in a former 

 memoir as being incongruously true. No further step of im- 

 portance towards 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 en- 

 riched it with new views and applications ; and through the 

 labours of ihese 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 interesting departments of modern 

 algebra. 



It must, however, be owned that some obscurity seems 

 still to hang over the subject, and that a further examination 

 of its principles may not be useless or unnecessary. 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 oflered itself 

 originally as the subject of operations unknown and undctcr- 



