of Fourier s Theorem. 283 



2 ( 2n 

 A i m ... = TaZS* f(*,fi,7 • . .) cos (la + w0 + ny + ... ) eM/3i 7 . 



W Jo .... (8) 



B / OT ... = 7^Z^ f(a,/3,y...)sm{la + m/3 + ny+ ,..)dadj3dy. 

 ^ ) Jo .... (9) 



These values of the coefficients may also be obtained directly 

 if we assume the expansion to be expressed in the form (5). 

 For if we multiply both sides of (5) by cos (lx + my + . . . ), 

 and integrate between the limits zero and 27r with respect to 

 each variable, we obtain at once 



J, 



f(x,y, . . .) cos [lx + my + ...)dxdy . . .= ±A(27r) n , 



which agrees with (8). 



When the variables are taken to be the coordinates on, y, z 

 of a point in space and the time t, we have the form 



f(x, y, z,t) = ^A cos ( px + qy + rz + st) 

 + 2 B sin (px + qy + rz -f st), 

 or the equivalent form 



fix, y, z, t) = 2a cos (px +qy + rz + st + a), 



where a and a are constants expressible in terms of the defi- 

 nite integrals A and B by the equations 



a = (A 2 + B 2 )% tan a = - B/A. 



This, then, is the analytical expression of the general theorem 

 enunciated by Dr. Stoney, and means that any disturbance 

 whatsoever in a medium either uniform or heterogeneous can 

 be resolved into systems of trains of plane waves. 



That some such resolution should be possible is not sur- 

 prising when the matter is regarded from the physical stand- 

 point. For two trains of plane waves traversing space in 

 different directions will produce illumination of a simple gra- 

 ting type on a screen placed perpendicular to the plane of the 

 wave normals ; that is, the pattern on the screen consists of 

 parallel bars, and the amplitude as we pass across a bar varies 

 according to the cosine law. If therefore any pattern, how- 

 ever complex, can be built up by superposing various simple 

 grating patterns, it follows that any disturbance within a given 

 region can be resolved into plane waves, and that any function 

 of x, y, Zj t can be resolved into a sum of cosines or sines of 



