310 Cambridge Philosophical Society. 



amining the coefficient of sin in the wth term. In a similar man- 

 as 



ner the discontinuity of f{x) or any of its derivatives may be ascer- 

 tained, and the amount of the sudden change of the function deter- 

 mined. In such cases the expansions of the derivatives oif(x) can- 

 not be obtained by differentiating under the sign of summation, but 

 are given by formulas which the author has considered. 



The most important case in considering a series of sines, is that 

 in which /(a?) is continuous ; but f{6) and/(«), instead of being equal 

 to zero, are given quantities, and the coefficients in the expansion 

 are indsterminate. In this case the coefficients in the expansions of 

 f'(x) and /"(a:) contain, in addition to the indeterminate coefficients 

 which enter into the expansion of/(x), the given quantities /(o) and 

 /(a). Thus the expansion in a series of sines is useful, not only when 

 f(o) and /(a) vanish, but also when they are given quantities. In the 

 same way the expansion oif(x) in a series of cosines is useful when/' (0) 

 and/*(a) are given, as well as when they vanish. Thus, to take an 

 example, the permanent temperature in a rectangular parallelepiped, 

 when the temperatures of the faces are any arbitrary functions of 

 the co-ordinates, can be expressed in a double series of sines in- 

 volving any two of the three co-ordinates. 



The author has only considered a series of sines and a series of 

 cosines, with the corresponding integrals ; but the methods which 

 he has employed are of very general application. The comparison 

 of different expressions of the same function of two or more inde- 

 pendent variables often leads to very remarkable formula?. The de- 

 velopment of arbitrary functions in the way considered by the author 

 is, however, not only curious but useful ; for the expressions thus 

 obtained are often much better adapted to numerical computation 

 than those which would be obtained by the developments usually 

 employed. 



In connexion with these investigations, the author was led to con- 

 sider the discontinuity of the sums of infinite series, or of the values 

 of integrals between infinite limits, which sometimes takes place even 

 when the series or integral remains convergent, and the general term 

 of the series, or the quantity under the integral sine, is a continuous 

 function of some quantity which is regarded as variable. The 

 author has shown that in all such cases the convergency of the series 

 or integral becomes infinitely slow. 



The problem of determining the potential due to a given electrical 

 point within a hollow conducting rectangular parallelepiped, and 

 to the electricity included on the surface, is solved by a method 

 which leads very readily to the result. The author thinks that a 

 similar method may sometimes be advantageously employed in other 

 questions. The electricity is first supposed to be diffused over a 

 finite space : this allows of the expansion of the potential V in a 

 triple series of sines. Instead of the equation y V = 0, where \jl 

 means the same as 



dx* df dz*' 



