Ork — Extensions of Fourier'' s and the liessd- Fourier Theorems. 13 



Mathematics, especially as the type of problem which suggested them can, 

 as iudieated, be solved without them by the aid of Love's general functional 

 solutions of the differential equation * 



Such expansions may, however, be used in proljlems of other types than 

 those which suggested them. For example, Lord Eayleigh, to illustrate the 

 effect of a yielding of the points of support of a vibrating string, has discussedf 

 its motion on the supposition that each end is attached to a massive particle 

 which is urged by a spring towards the position of equilibrium. He obtained, 

 without formally establishing its A'alidity, the expansion which is necessary in 

 the resolution of the most general motion possible into the fundamental oscil- 

 lations ; this expansion is an example, slightly more difficult than that just 

 alluded to, of the development in Trigonometrical Series which I established. 

 So, too, if the single particle at each end be replaced by a number elastically 

 connected, to one or more of which the string is attached, and if the motion 

 be investigated in a similar manner, still more comphcated examples present 

 themselves. , 



Moreover, in problems of this type, the boundary conditions might be such 

 that, of the type-solutions of the differential equations, the two which contain 

 respecti^•ely t f '■^'' and c"''^'' invoh'e different functions of x, and thus cannot be 

 combined so as to contain the skie or the cosine function alone of the time : 

 this would, for instance, be the case, if the particle or particles were subject to 

 frictional forces proportional to the actual or to the relati\'e velocities. Thus, 

 in such a case, the expansions in my paper would appear not to be precisely 

 those which are requisite. 



And in proceeding to remedy this defect I noticed, much to my regret, that 

 the expansions which I gave are so far VMnting in uniqueness and generality 

 that they mahe reference to only OTie arbitrary functio7i, whereas they nuiy he 

 required to satisfy conditions vjhich explicitly refer to tico.% In this connexion 

 it is to be borne in mind that, in many cases in which the characteristic 

 values of A can be grouped in pairs, equal in magnitude and opposite in sign, 

 the series of the former paper includes each term twice, whereas the usual 

 statements of expansions of Fourier type include each term once only. 



• Love, Phil. Trans., cxcvii., 1901; see also Lord Kelvin, "Baltimore Lectures," p. 193. 

 When it is desired to follow the history of a given disturbance in any medium, a solution which is 

 derived from the general functional solution, wiienever obtainable, of the difl'erential equation, what- 

 ever it may be, seems usually preferable to one of Fourier type. Moreover, as will be seen below, 

 it is a matter of considemble difficulty to justify formally the solution which the jiMJi-Fourier 

 treatment gives. 



t ''Theory of Sound," § 135. 



J c denotes the velocity of propagation, and t the time. 



§ In the solution of problems relative to Conduction of Heat, it does not seem that such an 

 extension is likely to be wanted. 



