284 Mr. T. Preston on the General Extension 



linear functions of the variables. It is not difficult to admit 

 that a very complex pattern may be built up by superposing 

 simple-grating patterns, but that any arbitrary pattern what- 

 soever may be so constituted requires demonstration. 



When the disturbance is periodic throughout space the 

 expansion in cosines or sines will represent the function com- 

 pletely throughout all space, and when the function is not 

 periodic the expansion can be made to agree with the function 

 at all points within assigned limits. 



In the case of a periodic function the expansion may be 

 derived from the condition of periodicity. Thus if X, fi, v, . . . 

 be the periodic intervals and /, m, w, . . . any whole numbers, 

 we have by the condition of periodicity, 



f(x,y,z,...)=f(ai + l\ y+mp, z + nv,...). . . (10) 

 If we denote the operator — by D 1; and — by D 2 , &c, we 



CLlL Ct-t/ 



have by applying Taylor's theorem to (10), 



fa, y, z, . .. . )= e™' +»<^ + *</(*, y, z, . . . ), 



■j- j-io-j- "JO « 



( 6 W>i+«Ml>l+...-H/(j.,y^ J ...)=0. ... (11) 



Now any differential equation of the form 



F(D 1 ,D 2 ,D 3 ,...)/K2/,i,...)=0 . . . (12) 



will be satisfied \>y f =e ax]rby+cz+ ' % ' provided a, b, c, &c. satisfy 

 the equation 



F(a, b,c,...)=0 (13) 



Hence equation (11) will be satisfied by a sum of terms of the 

 form 



fa,y,z,...) = ZAe°* +b " + °* + -, 



provided a, b, c satisfy the equation 



That is a,b,c,... must satisfy the equation 



l\a + m/jub + nvc + . . . = 2/r7r, 



where r is any whole number. Thus in the case of three 



* This method was employed by Mr. J. O Kinealy, for the case of a 

 single variable (Phil. Mag. August 1874, pp. 95, 96.) 



