in the Position of a Point in Space. 133 



== — Wi a^ — Ml • - - . 



(46) 



The middle coefficient of this block is l;,j,k-, and its coordinates are 



r=^ //..■, y=iz^2/, 2=^-//2. (47) 



When the n is made an infinity of the second order, and the coeffi- 

 cients in the expansion are set close together so as to be consecutive, 

 the expansion extends throughout infinite space, the intervals J;>', 

 Jy, Az become <fo;, dy^ dz, the coefficient li^j^k becomes the function w 

 which represents the limiting form of the series of coefficients I, its 

 coordinates become 



.r=ul'; y=Jdy, z=kdz, (48) 



and we wish to express w as a function of x, y, z. For this purpose, 

 we regard the terms in the block as forming an algebraic triple 

 series, whose first and second differences ai"e to be taken into account, 

 and the differences used are to be symmetrical. 



The formula for symmetrical finite differences, where ti is a func- 

 tion of the three variables a/ix, hAy and cAz, is found in a manner 

 quite similar to that employed in obtaining the formula (5) for two 

 variables. The coefficients of the differences in the x direction are 

 like those in (3), while those in the y and z directions are the same, 

 only writing h and c instead of a. Then for some other difference, 

 Z/g 3^ for instance, the coefficient will be the product of the coeffi- 

 cients of zJg ,j.o, J„.3.„ and z/o.,,.^. The method of forming the differ- 

 ences so as to keep u^.^.y^ always in the middle, is analogous to that 

 already explained in discussing formula (5). The formula now 

 obtained, if we stop with second differences, will be 



a 



b . c 



W«,Ae = Wo.O.O+-j-^tO.O +^^0.1.0 +-f ^0.0.1 



a^ h^ G^ A 



+ ^^2.00 +^-^0-2.0 +]~^^0-0.2 



ah , ac . Jc . /.^x 



Let the first and second differentials of w in the cc, y and z direc- 

 tions be written here for the corresponding differences A. Then any 



