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XXXIX. On the General Extension of Fourier s Theorem, 

 By Thomas Preston, M.A., F.RM.1* 



IN a recent number of the Philosophical Magazine Dr. G. 

 J. Stoney | has announced the following important optical 

 theorem in relation to microscopic vision: — " However complex 

 the contents of the objective field, and whether it or parts of it be 

 self-luminous or illuminated, in any way, however special, the 

 light which emanates from it may be resolved into undulations 

 each of which consists of uniform plane waves." 



In this theorem it is explicitly stated as a general principle 

 that any disturbance however complex within a given region 

 of space may be resolved into a system of plane wave com- 

 ponents, or that a system of trains of plane waves can be 

 determined such that when compounded they will reproduce 

 any arbitrarily specified disturbance throughout a given 

 region. 



When the disturbance is a function of a single variable this 

 statement forms the verbal expression of Fourier's Theorem, 

 viz. that any arbitrary function of a variable can be expressed 

 as a series of sines or cosines of multiples of that variable, 

 between assigned limits ; and when the disturbance is a func- 

 tion of two or more variables the theorem in its most general 

 form implies that any function of any number of variables 

 can be expanded in a series, each term of which can be ex- 

 pressed in terms of a linear function of the variables. 



Since the time of Fourier it has been customary to represent 

 a general function of any number of variables by a series each 

 term of which is a continued product of cosines of multiples 

 of the variables ; thus 



f(x, y } z) = 2 A cos Ix cos my cos nz. . . . (1) 



This form in fact was used by Fourier in his investigation of 

 " the movement of heat in a solid cube." Now if the expo- 

 nential values of cos Ix &c. be substituted for them, it is seen 

 at once that the product of the cosines cos Ix cos my cos nz 

 resolves itself into terms of the type cos (lx±my±nz), and 

 consequently the form of expansion (1) in the case of any 

 number of variables is equivalent to 



f(x, y.z, . . .) = 2JA cos (Ix + my + nz + ...); 



* Communicated bv the Author. 



f Phil. Mag. vol. xlii. p. 332 &c. (1896), and vol. xliii. p. 139 (1897). 



