BY FINITE DIFFEEENCES OF PHYSICAL PROBLEMS, ETC. 353 



second, with the same coefficient that the value of (f> at the second enters into the 

 body equation at the first. For example, if fa, fa, ... fa are spaced in order at 

 equal intervals along the X axis, and if 2) = d 2 /da?, then Q'fa = fa-i 2fa + fa+i 

 and (7) is satisfied when the (i)'s are simply unity. But if 35 = 3/9x, then 

 <)'fa = pSfa = fa^fa^, so that c a is +1, while c lk = 0, so it is impossible to find 

 a unifying factor. 



Next, as to ij^e^f^ = i^e kj f j{ . A. particular condition in which these sums become 

 identical is if f jt = e^ify where a, is an arbitrary number, independent of the (|/)'s. 



By (6) this equation implies that 



Equation (10) states that the boundary values will satisfy their part of (4) if every 

 boundary number is formed as follows : Take each body -point. Multiply the value 

 of <f> there by the unifying factor and by the coefficient of the said boundary number 

 in 2/< at this point. Sum this product for all body points and multiply the total by a 

 number independent of the body values. This is not the most general way of 

 satisfying (9), but it is a common one in physical problems. Thus, for example, if 



(-32 32 \ 



TT 5 + ^r~9 ) 4>v. 0, a boundary condition of frequent 

 ex - 



occurrence is to have fa given, and therefore fa, fa = 0. This corresponds to <*,- = 0. 



Another common condition is = -^ = &% $21 , where 8?^ is an element of the 



tin tin 



normal to the boundary. Then a boundary number will either be equal to a body- 

 number or to a weighted mean of two neighbouring body-numbers ; in either case 

 a," 1 = e l ji l + e 2j i ;t + ... + e nj i n , in which all but one or two of the (e)'s will vanish. 



Lastly, it will be necessary in what follows that V should be of one sign for all 

 values of fa, fa, ... fa. This will be so if 3 2 V/3</</ is of the same sign for every i//. 

 Whence, by (3), (5), and (6), i k {c kk + '2 l e k jf jk } must be of the same sign for 

 every k ........................ (11). 



As before, we will treat only the case in which the boundary and body equations 

 satisfy this separately, so that every i^c/^ and every i k ^,e ki fj k have all the same sign. 



For simplicity we will take V as positive ............ (12)- 



Then i k has the same sign as c kk . Further, as the sign of the operator 2)' does not 

 affect the solution of the equations tyfa = 0...<>'fa = we may alter it arbitrarily, and 

 we will for simplicity suppose it chosen at each point so as to make e w positive . (13). 



The ii...i n will then also be positive .............. (14). 



For convenience, a table is added giving the forms of I and V for some common 

 forma of 2)'. These were taken by analogy from infinitesimals, and afterwards 

 verified by trial. Thus for 3) = V 2 we have V = J JJJ [V<] 2 dr, where V is the vector 

 operator Nabla and dr an element of volume. On expressing V in orthogonal curvilinear 

 co-ordinates qiq a q 3 and varying fa it is found that SV = JfJ S</> . V 2 < . I . dr + a surface 



VOL. COX. A. 2 Z 



