522 Transactions of the Society. 



dark ring subtends the same angle at the centre of the object- 

 glass, as a single wave-length does at a distance equal to the 

 length of a side of the square aperture of the object-glass. 



The length of the radius of the first dark ring in a telescope is 



cXf 

 determined by the formula — — , where c is a coefficient, X the 



wave-length, / the focus, and a the aperture of the object-glass. 



The value of X for white light will, throughout this paper, be that 



suggested by Mr. J. W. Gifford, viz. ^-g^-gj inch = 5607 tenth 



metres. 



As the radius of the first dark ring subtends an angle at the 



c\ 

 centre of the object-glass of — , and as the limit for the resolving 



power of a telescope is required in this angular measurement, the 

 quantity / may for the present be disregarded. Lastly, we come 

 to the coefficient c ; the experimental determination of this 

 quantity, both for the telescope and for the microscope, is the 

 subject with which we are concerned this evening. When the 

 aperture is square, the value of c is unity, and so the formula just 



given above may be written , which expressed as an angle is 



4" "55. Theoretically, then, a telescope, with a square aperture, 

 will separate any double star whose components subtend an angle 

 of 4" • 55 divided by the aperture of the telescope in inches. Now, 

 as the object-glasses of telescopes are circular, and not square, 

 mathematicians have integrated this quantity for a circular aper- 

 ture, and have found the value of c to be 1*2197; therefore the 

 theoretical separating limit for a circular aperture, when expressed as 

 an angle, is 5" " 55, or one second more than that for a square aperture. 



We now come to the unsatisfactory part of the subject, viz. 

 the practical separating limit for a telescope with a circular 

 aperture is the same as the theoretical limit for one with a square 

 aperture. In other words, the theoretical limit is 18 per cent, 

 greater than it ought to be ! The theorists say that the quantity 

 determined is the radius of the first dark ring, which may not 

 necessarily be the separating limit. This is true, but if the radius 

 of the first dark ring be experimentally determined, it will be found 

 to be 32 p.c. smaller than its theoretical value. What would 

 be thought of the Newtonian theory of gravitation if the calculated 

 distance of Mars from the Sun was 32 p.c. too great ? The 

 theoretical value of c for the measurement of the first dark ring 

 with a telescope having a circular aperture is, as stated above, 

 1*2197, while the value as experimentally determined is 0*8266, 

 or 32 p.c. less. 



The empirical formula for the telescope limit employed by a 

 most expert double star observer, the late Eev. W. K. Dawes, 



