BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 357 



it follows that when f=P k then V/T = X* 2 , and when f differs slightly from P* then 

 V/T will differ from X* 3 by a small quantity of the second order (RAYLEIGH, ' Theory 

 of Sound,' 88). In 3 '2 '3 a guess is made at P k , and X A 2 is then calculated roughly 

 as V/T. 



The special values X^.-.X,, 3 for any region are included between the greatest and 

 least of the X 2 pertaining to any region, which includes the former region when the 

 boundary values of both regions vanish. For let P t , the integral of ($)' + X* 2 ) P t = 

 for the smaller region, be set round about with noughts until a larger boundary is 

 reached. The conditions for the existence of a function V will then be satisfied for 

 the larger region. Let quantities belonging to the larger region be distinguished by 

 dashes. Then by (28) P A may be expanded in the form 



P A = SBjP';, where the (B)'s are constants. 



Now by (31) X/ = V/T, when the body values have the values given by $ = A A .P A .. 



Also, by (15), T' = T since the added squares are zero. Also, by (3), i^'\\> k is equal 

 to both 3V/3i// A and BV'/S^ so that V and V can only differ by terms independent of 

 //!, i// 2 , ... />. But there are 110 other variables in V. Therefore V EE V and X A 2 = V'/T'. 



Now, expanding V and T' by (17) and (18), 



3 _ X 1 / 'G 1 g +X a "G a 2 +...+X r /a G r il 

 G 1 f +Q f > +...+G/ 



And therefore X/ lies between the greatest and least of the X' 2 . If, however, the 

 boundary values do not vanish then 9V'/3^ = +4 < >'i// A: when the (ft)'s are regarded as 

 independent of the (i//)'s during the differentiation, whereas 3V/3*/;* = the same 

 quantity when V is expressed as a function of the (i/>)'s entirely, and a more detailed 

 examination shows that X* 2 need not lie between the greatest and least values of X' 2 

 for an enclosing boundary with zero boundary values. However, X t 2 is necessarily 

 finite, for it is the root of a rational integral function with finite coefficients. 



