of Fourier's Theorem. 285 



variables, the quantities ia, ib, ic may be the coordinates of 

 any point on any one of the system of planes 



/\£ + m/AT) -f nv£= 2rir, 



in which Z, m, n ) and r are any whole numbers, positive or 

 negative. The general expansion is consequently of the form 



/(#, y, z, . . . ) = Z A cos (ax -\-by + cz + ...). 



In the particular, or rather extreme, case in which the func- 

 tion /has a given value at each of a system of equally-spaced 

 points and has zero value at all other points of space, we have 

 the disturbance contemplated in the lemma used by Dr. Stoney 

 as the foundation of the demonstration sketched in his first 

 paper*. 



It is interesting to notice that any function of an odd 

 number of variables can be expanded in a series of sines or a 

 series of cosines of linear functions of multiples of the variables, 

 as in the case of a single variable, the limits of integration 

 being zero and ir. On the other hand, when there is an even 

 number of variables the limits of integration require to be 

 zero and 2tt. 



In the case in which I, m, n, &c. are not whole numbers the 

 function may still be expanded in the form of a series of sines, 

 or a series of cosines ; thus, 



f (x, ij ...) = 2 Asm (Ix + my + .. .), 

 where /, m } &c. are chosen so as to satisfy an equation of the 



cos 

 sin 



{ (I + V) x + (m + m')y. . . } ™* { (I - l')x + (m - m')y. . . j 



Q + l'){m + m') ... {I — I') (m-m r ) . . . 



and the coefficients A &c. are determined by definite integrals 



of the form (8) or (9). In the case of a single variable this 



equation reduces to 



tan Ix 

 — = constant, 



Ix ' 



the form obtained by Fourier. 



* Loc. cit. 



