248 Prof. Everett on the Optics of Mirage. 



and in the limit, when x = r, 



Dg _ 2b-\rr+g 



D 1 ~^ X b{r+g)+2gr 



When r=g, =^ = 1, showing that the two methods are equally 



1 



sensitive for that value of r or x which equals the resistance of 



one coil of the galvanometer. When r is greater than g, the 

 ordinary method is to be preferred, for ~ is then less than unity. 



It can, however, never be less than h 9 , which happens when 

 r is infinite. 



But for values of r less than g, -^ is greater than unity, and 



\ 



increases rapidly as r is reduced, until in the limit, when r = 0, 



V 1 ^ b 



This proves that when the resistance to be measured is smaller 

 than that of the gaivanorneter-coil, the second method is much 

 to be preferred. For instance, let the battery have a resistance 

 of 10 ohms, the galvanometer (each coil) 500 ohms, and r= 10 

 ohms, then the second method is 17 times as delicate as the first; 

 and if r were 1 ohm, the second method would be 416 times as 

 delicate. 



In fact, if, after getting as true a zero as possible by the ordi- 

 nary method, the connexions be altered to the second arrange- 

 ment, the slight inequality between r and x, which was inappre- 

 ciable by the ordinary method, will be at once rendered evident 

 by a large deflection of the needle. 



XXXII. On the Optics of Mirage. (Second Paper.) By Professor 

 Everett, M,A., D.C.L., Queen's College, Belfast*. 



XL W/'E proved in the preceding paper that wherever a plane 

 T of maximum index exists, the surfaces-of-equal-index 

 being parallel planes, the law of index- variation in its immediate 

 neighbourhood must in general be 



d\o%p _ y^ 

 dy « 2 



and that this law implies the existence of conjugate foci at the 

 mutual distance ira. 



It may be proved in like manner that, when the surfaces-of- 



* Communicated by the Author. 



