detect Aberrations of the Second Degree, d73 



which interests us. If we could measure the perpendicular 

 distance between these wave-fronts at a sufficient distance 

 from B, we should know the angle between them. But h" 

 is only a fictitious line. What we cannot measure between 

 h and h" we can measure between a and a! } provided we can 

 determine the point of intersection between a and a!, and 

 provided this be found in a convenient position. AVe have 

 therefore to determine the point of intersection of a and a', 

 knowing that of h and h". 



The observing telescope is shown at T, fig. 5. Its axis is 

 parallel to B II. "We will show that the phase-difference of 

 >i and a' is constant at all points on any line parallel to the 

 line B II. or to the axis of the telescope. 



If we write X. A/, not for icave-lengths, but for the per- 

 pendicular distance between consecutive wave-fronts of the 

 same phase, and £, & for the total aberration of the wave- 



X f X 

 fronts of the two svstems, we have to show that ^ — ^ 



J COS COS 6 



is identically zero for eight specified equidistant azimuths, 



and is not greater than 0*3 ™ for other azimuths. Each 



of these quantities is determined by a complicated expression; 

 and the equality specified can be most readily determined by 

 trigonometric computation. 



To prove the proposition, therefore, we will take that 

 azimuth where, according to Dr. Hicks, the shifting of the 

 intersection is a maximum, and we will assume the extreme 

 case where the velocity of the apparatus is half that of light. 



In fig. 6, the mirrors D, I, and II are accordingly supposed 

 to move in the direction of the arrow. Let t be the period 

 of the waves of light incident on D ; according to the 

 previous specification, the angle between these wave-fronts 



and the plane of II is sin -1 ^ cos a. ; that is, they are parallel 



to II. Lay off on e d r the line of motion of a certain point 

 of the mirror D, the positions of this point at the times 0, t, 

 'It. &c. Positions of D and of I and II at certain times are 

 also noted in the same way : all numerical subscripts denote 

 times. The source moves with the apparatus, and therefore, 

 with the assumed ratio of velocities, the apparent wave- 

 length of the light incident at D is half the wave-length in 



DO o 



the case of rest, and i< half the distance described by a 

 -front in the unit of time. Let the initial position be 

 one in which a wave-front passes through the given point in 

 me mirror and through the point in the line of motion. 

 \t the time t — r the mirror is at 1, and the wave-£ront in 



