in Prof. Rowland's Concave Gratings. 195 



perfectly explained through the uncertainty in the two 

 adjustments of the centre of curvature of the mirror on the 

 axis of the carriage of the eyepiece, first in the direction of 

 the girder that unites the two carriages, and secondly, in 

 the lateral direction. Of this I have convinced myself by 

 another series of determinations, by displacing intentionally 

 the centre of curvature. A fault of 0*5 millim. in the deter- 

 mination of the radius of curvature is sufficient to explain the 

 before-mentioned difference. 



Thus the point 143*8 is to be considered as the vertex of the 

 right angle of the rails, through which passes in the present 

 case the theoretic focal circle of the grating, or rather a 

 curve which differs from it very slightly. Using this 

 number (a ) I have calculated the differences d l = a 1 — a and 

 d 2 = a 2 — a Q , which are found in the table under the heading 

 " observed." A glance at these numbers shows that they 

 are at least very nearly proportional to the corresponding 

 values of nX, which implies that the angle e of the segment of 

 the true focal curve is constant. To examine this more 

 closely, we will insert in the preceding equation of e 



p cot e 



1 = x, 



CO 



and we will calculate by the method of least squares the exact 

 value of x from the 16 equations of the form 



x . nX = d, 



which we obtain from the preceding table on using all the 

 values d 1 and d 2 . 



In this way we find the value 



pcote , Q „., , Qn 



x = - = i8261-|-80. 



co — 



The numerically equal values of d 1 and d 2 , which are 

 obtained on making use of this value of x, are given under 

 d in the last column of the table. The differences between 

 these numbers and the observed values being confined within 

 the limits of errors of observation, it must be considered as 

 proved that the angle e in the segment of the focal curve is a 

 constant. 



A segment of which the angle is a constant belonging 

 necessarily to a circle, we can express the result of our re- 

 searches as follows: — 



The focal curve which passes through the centre of curvature 

 of the mirror is a circle, which, however, has not the radius of 

 curvature in the apex of the grating as a diameter. 



