BY FINITE DIFFEEENCES OF PHYSICAL PEOBLEMS, ETC. 323 



appears in the table. The intermediate stages may be wildly irregular if the values 

 of a decided upon are used in ascending order of magnitude. If, on the contrary, 

 the descending order is pursued the table tends to improve more regularly. 



3 '2 '2. The Equation 1)'< = p, ivhere p is a completely known Function of 

 Position This can be treated in the same way as $)'< = 0, except that the 

 approximations must now be of the form 



<A.+i = 4>-*~~ l< S>'4>-p ......... (1), 



for, since ^)'<,, = />, this may be written 



<^. + i-<^ = < M - </>,,- a" 1 $>'(</>,-</>,,) ....... (2), 



which is the same as (3) of 3'2'L 

 3-2-3. The equation 



($'-X 2 )P = .......... (1), 



together with s homogeneous boundary equations such as 



Pi = 



where ft^ ... ft, are the boundary values, i//, ... i/;,, the body values and the (/)'s are 

 given numbers. P is now written in place of <, because by (4) of 3'2-l P is defined 

 as satisfying equation (1) of this section. We will suppose that both P and X 2 have 

 to be determined. As equations (2) contain no terms independent of the (i/)'s, it 

 follows that when these expressions for the (/3)'s are substituted in the body-equations 

 the latter become homogeneous, and are only consistent for the particular values of 

 X 2 which we have already denoted by \\, X/, ... X n 2 . Further, on account of this 

 homogeneity, any multiple of an integral satisfies the correct boundary conditions. 

 Such a possibility does not arise with $)'< = 0, for as I)' contains no adjustable 

 constant such as X 2 it has no integral save <f) = 0, unless the boundary equations 

 contain a term f jt> independent of the (/)'s. And if they do contain such a term, any 

 multiple of an integral fails to satisfy them. Thus, in the theory of membranes, if we 

 put 1) = &/Ba? +&/&/*, and (j) for the small displacement of the membrane from a 

 fixed plane, then in the case of the membrane at rest 5)'< = 0, and <j> is commonly 

 given at the boundary, its values there being f 10 , f x , ... /,. 



On the other hand, for the vibrations of the membrane ()'+X 2 ) < = the boundary 

 condition is commonly <f> = 0, corresponding to the vanishing of all the (/)'s. 



To return to the general form of $)' : let P* be the integral desired ; (P^, (P*) 2 , . . . 

 (P A )<+i the initial guess and successive approximations to P*. The fact that any 

 multiple of an integral is itself an integral allows us to put for the body-points of 



(P*)m+l 



(P,) m+1 = 7 (P,) m -a- 1 D'(P,) m ....... (3), 



2 T 2 



