36 M. G. Kirchhoff on the Relation of the Lateral Contraction 

 hence 



A' -a' 



7' 

 ff 



B'-Z/=-^-c\ 



r 



But since the line N£ lies in the plane of the triangle OSf, and 

 bisects its angle at f, we have 



Y'-b'=(B'~b') 



_V!_ 



2 7 ' 2 -i 



From which there follows, 



X'-a'=-a' 



_2y_ 



2 7 ' 2 -l 



Y'-6' = 



^3U 



(4) 



2V 2 -1 

 By a corresponding notation we obtain for the second mirror, 



X"-a"=-a" 



2 7 " 



2 7 " 2 -l 



Y'/-^-^'. 



27" 



2 7 " 2 -l 



^ 



c". 



(5) 



If (n'#), (n f y), (n'z) are the angles which n f the normal to the 

 mirror forms with the axes of x, y, z proceeding from A', we 

 have 



a' = cos (n'x) + u' } cos (n'y) + a ; 2 cos (n'z), 



0=fi' o cos (w'#) + cos (n'y) + /3' 2 cos (n'z). 



a\ or — -/3' is the angle by which the rod has twisted about a 

 vertical axis from the position in which its axis is parallel to the 

 f axis; this angle, if not =0, is certainly very small; and since 

 cos (rHy) and cos (n'x) are also only small magnitudes, we may 

 put 



«' = cos (n'x) + a' 2 cos (n'z), 



= cos (n'y) + /3' 2 cos (n'z) . 



In order to designate the values which the magnitudes under 

 consideration assume in the case in which the rod is straightened 

 in the manner indicated above, I shall place a — over them. In 

 that case 



cos n(n'x) = a', cos (n'y) = /3', cos (n'z) = 7'. 



