534 Mr. stokes, ON THE CRITICAL VALUES OF 



I have next shewn how the existence and nature of the discontinuity of /'(a') and its derivatives 

 may be ascertained merely from the series, whether of sines or cosines, in which f{x) is developed, 

 even though the summation of the series cannot be effected. I have also given formulas for 

 obtaining the developenients of the derivatives of /(.r) from that of /(«) itself. These develope- 

 ments cannot in general be obtained by the immediate differentiation of the several terms of the 

 developement of f(v), or in other words by differentiating under the sign of summation. 



It is usual to restrict the expanded function to be finite. This restriction however is not 

 necesssarv, as is shewn towards the end of the section. It is sufficient that the integral of the 

 function be finite. 



Section II. contains formula? applicable to the integrals which replace the series considered 

 in Section I. when the extent a of the variable throughout which the function is considered is 

 supposed to become infinite. 



Section III. contains some general considerations respecting series and integrals, with reference 

 especially to the discontinuity of the functions which they express. Some of the results obtained 

 in this section are referred to by anticipation in Sections I. and II. They could not well be 

 introduced in their place without too much interrupting the continuity of the subject. 



Section IV. consists of examples of the application of the preceding results. These examples 

 are all taken from physical problems, which in fact afford the best illustrations of the application 

 of periodic series and integrals. Some of the problems considered are interesting on their own 

 account, others, only as applications of mathematical processes. It would be unnecessary here to 

 enumerate these problems, which will be found in their proper place. It will be sufficient to 

 make one or two reiiiarks. 



The problem considered in Art. 52., which is that of determining the potential due to an 

 electrical point in the interior of a hollow conducting rectangular parallelepiped, and to the elec- 

 tricity induced on the surface, is given more for the sake of the artifice by which it is solved than 

 as illustrating the methods of this paper. The more obvious mode of solving this problem would 

 lead to a very complicated result. 



The problem solved in Art. 54. affords perhaps the best example of the utility of the 

 methods given in this paper. The problem consists in determining the motion of a fluid within 

 the sector of a cylinder, which is made to oscillate about its axis, or a line parallel to its axis. 

 The expression for the moment of inertia of the fluid which would be obtained by the methods 

 generally employed in the solution of such problems is a definite integral, the numerical calculation 

 of which would be very laborious ; whereas the expression obtained by the method of this paper 

 is an infinite series, which may be summed, to a sufficient degree of approximation, without much 

 trouble. 



The series for the developement of an arbitrary function considered in this paper are two, a 

 series of sines and a series of cosines, together with the corresponding integrals ; but similar 

 methods may be applied in other cases. I believe that the following statement will be found to 

 embrace the cases to which the method will apply. 



Let w be a continuous function of any number of independent variables, which is considered 

 for values of the variables lying within certain limits. For facility of explanation, suppose u a 

 function of the rectangular co-ordinates x, y, z, or of x, y, z and t, where t is the time, and 

 suppose that ii is considered for values of ,r, y, x, t lying between and a, and b, and c, 

 and T, respectively. For such values suppose that u satisfies a linear partial differential equation, 

 and suppose it to satisfy certain linear equations of condition for the limiting values of the 

 variables. Let f7 = 0, f/' = be two of the equations of condition, corresponding to the two 

 limiting values of one of the variables, as x. Then the expansion of « to which these equations 

 lead may be applied to the more general problem which leads to the corresponding equations of 

 condition U = F, U' = F', where F and F' are any functions of all the variables except x, or of 

 any number of them. 



