282 Mr. T. Preston on the General Extension 



which is the analytical expression of the foregoing general 

 statement when the variables are taken to be x, y, z, t, namely 

 the coordinates of a point in space and the time t. 



In order to investigate this expansion we return to the case 

 of a single variable and write Fourier's expansion in the form 



wf(x) = y, cos Lv I f(a) cos a da + 2 sin Lv f(a) sin a da 



Jo J 



= 2 f^/W cob /(*-.«)&, (2) 



where I is any whole number positive or negative. 



Now, if in (2) we replace f(x) by a function of two variables 

 f(x, y) , we have 



C 2rr . 



T/C^j 3/) = 2 ) ffay) cosl(x— a) da ; . . (3) 



Jo 



and if /(a, y) in the right-hand member of (3) be replaced by 

 its expansion in a Fourier series of cosines and sines of mul- 

 tiples of y, as denoted in equation (2), we have 



ir 2 f{x s y)= 1 2% | I /(*,£) cos I (w— a) cos m{y-/3)d*d/3 



Jo Jo ' . .' . . (4) 



where I and m are any whole numbers. Similarly the equa- 

 tion for any number of variables may be written down at once. 

 Since the general term of (4) is of the form Ae l{lx+m ^\ it is 

 clear that any function of any number of variables can be 

 expanded in the form 



f(xj y, . . .)=S A cos (Lv + my + ...) + % B sin (Lv +my + . . .) ; 



.... (5) 



where I, m, &c. are any whole numbers positive or negative? 

 and the coefficients A, B, &c. are given by (4). 



Thus, in the case of two variables the coefficients of 

 cos (Ix + my) and sin (Lv + my) are respectively 



A/. = -p— 2 J /(*, 0) COS (lu + m/3) d*d/3, . . (6) 



O /~2tt f* lit 



Bfa=75%| f{*,P)sto<}*+mftdadfr . . (7) 



( 27r ) Jo Jo 



and similarlv in the case of n variables we have 



