196 Mr. G. Pierce on Indices of 



This is a solution of the equation of condition for maximum 

 or minimum, with an error in cos 6 not greater than *0013. 



cos = ±(1- -0013), 

 6 =7i7r + -05, 



-— =?27T+ # 05, 

 A, 



rik '05\ 



= — +'0o5 centim. 



z 



The last term, making a difference of less than I millim. in 



the value of ~, is negligible. 



Coming now to the general case, let us assume f(r) any 

 function of r, analytic everywhere outside of a small cylinder 

 about the oscillator, and of which all the derivatives exist. 

 Let f{r) vanish at infinity to such an order that r ,f(r) is 

 finite — an assumption suggested by the fact that the rate of 

 propagation of energy through any closed surface about the 

 oscillator is finite. Then we can expand f(r) in the form of 

 a descending power series with undetermined coefficients, 

 thus 



Differentiating term by term (assuming the proper con- 

 vergence of the series), we have 



Whence 



<jA— - 



Ci 2c 2 3c 3 



1 )— - 



A — 



rp'v rviO n-A 



X r 2 r' d r A 





tlT £l + C A + e A 



where r is written for (a ■+■%). 



Whether or not this expression squared is negligible in 

 comparison with unity depends on the values of the co- 

 efficients c. Theoretically, f(r) can be obtained as an 

 integral, from Hertz's equations and the elementary theory 



