308 ME. L. F. EICHAEDSON: APPEOXIMATE AEITHMETICAL SOLUTION 



methods of successive approximation apply. But, of course, the labour would be 

 greatly increased. 



I'l. Finite differences have, in themselves, but little importance to the student of 

 matter and ether. They are here regarded simply as a makeshift for infinitesimals ; 

 and the understanding is always that we will eventually make the differences so 

 small that the errors due to their finite size will be less than the errors of experiment 

 or practical working, and may therefore be disregarded. That it is possible to make 

 them small enough without much labour is illustrated by examples given hereafter. 



In consequence of this point of view, the notation employed for finite differences is 

 very similar to that for infinitesimal differences. Thus d and 3 are differential 

 operators, while 8 and $ are the corresponding finite difference operators. The oft- 



3 2 3 2 3 2 

 occurring symbol V 2 = - 2 + ^-^ + 2 is represented in finite differences by ^ 2 . 



The differences employed are " central differences," that is to say, they are 

 considered as existing at the centre of the group of co-ordinate points from which 

 they are derived. In this respect the notation differs from that used and defended 

 by BOOLE (' Calculus of Finite Differences,' Art. 14), in which A n x is considered to 

 exist at one end of the set of n+l quantities which contribute to its value. The 

 differencing operator 8 and SHEPPARD'S* averager /A are defined by 



Bf(x) =f(x + &)-f(x-W ........ (1), 



........ (2), 



where li is the co-ordinate difference. SHEPPARD shows that /JL, 8, and d combine with 

 one another according to the ordinary rules of algebra. 



In this paper the differential coefficient d"f(x)/dx" will be approximately represented 

 when u is even by h~ n .B*f.(x) at the tabular points, and by h~ n .p.b n f(x) half-way 

 between the tabular points. That is to say, these difference ratios are taken in place 

 of the differential coefficients, and the error caused by so doing is left for consideration 

 until after the difference equation has been solved. When n is odd, the symbolic 

 expressions given above for d*f(x)/dai? at tabular and at half-way points are simply 

 interchanged. The representation is closer when the averager p need not be intro- 

 duced. Partial differential coefficients are represented by the difference ratios found by 

 performing the above operations with p. and 8 for each independent variable in turn. 

 It will be convenient to have the representation of some of the commonest differential 

 coefficients set forth explicitly. 



Let (j> be a function of x and y, and let lines be ruled on the plane xy parallel to 

 the axes at equal distances h, of x and y, so as to divide the surface into a number of 

 equal squares each of side h units. Let the arithmetical value of < at the centre 

 point of each square be written down in the square, forming a table of double entry. 



* W. F. SHEPPARD, " Central-Difference Formula," ' Proc. Lond. Math. Soc.,' vol. xxxi., p. 460. 



