4 M; n;or. A. c. DIXON* ON STIMIM uoi'vn.u-: HARMONIC KXI-ANSIONS. 

 Then, by help of the inequality, 



it follows that 



In the same way, if 



\\fx)'dx 



exists, <{> may be so determined that 



f* 

 (fxfatfdK < e, 



Ja 



by help of the inequality 



7. Thus, for the purpose of approximation, we are to divide the domain of a- into 

 sub-intervals, all tending to zero, and in each sub-interval put for p a suitable mean 

 among its values over the sub-interval, which may be conveniently called a local 

 average of p and denoted by r. Suppose, then, that 



fx pz \7/ 



?-U dx, U (x, a) = U = 1 - dx, 



.a Jo 1' 



v (.r, ) = v = 1 + f*>-V dx, V (.<-, a) = V = - f - <1s. 



.'a " I' 



In each sub-interv r al u, v, U, V are solutions of the equation 



and are, therefore, of the form 



A exp.r v/ X + Bexp (x v\). 



A, B are constants through each sub-interval, but are changed at the passage from 

 one to another. It is of great importance to ascertain whether they can increase or 

 decrease indefinitely, and we shall now prove that they cannot if the total fluctuation 

 of log r is uniformly limited,* that is, if the total fluctuation is always less than u 

 fixed finite quantity at all stages of subdivision of the domain. 



8. At every internal point of a sub-interval v, V have differential coefficients, and 

 at the points of division their derivatives, upper and lower, on the same side 

 are equal. 



* This does not imply that the total fluctuation of log p must be limited. For instance, p may lie 1 at 

 all rational points and 2 at irrational points, then r = 2 everywhere. 



