376 Dr. Barton and Mr. Lownds on Reflexion and 



48. It may be noted here also in corroboration o£ these 

 results that, if we simplify to the case of a resistance-bridge 

 of resistance infinity or zero and omit all small terms and 

 corrections, we obtain Bjerknes' expression for interference*. 



Thus taking only the first terms of F, G, and H, omitting 

 their denominators and the factor e~ 2 ^, and writing A=l, 

 and a = or 77- , we have 



F+G + H = 2(l±<r M cos0), . • • (40) 



which is equivalent to Bjerknes' equation just quoted. 



49. If we now simplify equations (35) to (38) by neglect of 

 k* and divide by the corresponding equation (30) , we obtain 



d' ^ 



y=|=F + G + H, 



where 



1 k 



F =J^ ~ j)? 9 sin 2k, 



\ r ^ 72 - ^ (sin 2« + r 2 sin 2 7 ) I > 



> (±1) 



cos ^ — a k 



R = Zse~ ke ^^^^-^(sin0 + a-r 2 sin0 + a -2K) 

 1 — r* \j 



J 



This is the general formula to the desiiled degree of 

 accuracy. 



50. Reflected System. — In order to realize an experimental 

 arrangement by which the value of A, the amplitude factor 

 on reflexion at the condenser, may be inferred as simply as 

 possible, let l 2 be taken very large. The term II is then 

 practically extinguished by its very small exponential factor. 

 But, lest its value should still be appreciable, let two values 

 of l 2 be used, viz. : / 2 + V^ anc ^ h~ V*S where \ is the 

 wave-length of the incident waves. These values of l 2 will 

 give to H opposite algebraical signs, and since H is here 

 very small the difference between its values for each position 

 will be quite negligible. Moreover the values of F and Gl- 

 are almost the same for / 2 + X/8 and / 2 — X/8. We may, 

 accordingly, write as the working formula for the reflected 

 system, 



mean y = F + G, (42) 



the l mean y ' in the experimental case being the arithmetic 



* Wied. Ann. xliv. pp. 513-526 (1891). Equation near top of page 

 517. 



