98 Mr. J. J. Guest on the Strength of 



hence 



^{0 2 (l + ^)-^i(l-^)}+y{^+^(l-^)-(l + ^-^ 2 -% 2 )} 

 + z{<j> l (l + 2 )-cf> 2 (20 1 + « 1 )}=Q; . (11) 



or to the first order, 



^ 2 -<j> 1 )+y{20 1 + cc-l-0 2 ) + z.<f> 1 =O; . . (11a) 

 therefore y is small, 



and x 2 + z 2 =l to the second order, 



and x + z = 1 to the first order. 



.*. w=l and z = 0. 

 [The solutions #=0, z = l refer to the point L.] 



.'. as before, x=\ to the second order. 

 From (10a) 1 + 02+* = 20 1 + a 1 -0 2 , 



.-. 3= 20,-202 + a; 

 and from (11a) <p 2 — <f> 1 —y = 0, 



••• y=<f>2-<l> l > 



Hence x=l+p, 



y = <p 2 — <pi + o; 



z = 2(0 1 -0 2 )+a + T, 



where p, a, and T are quantities of the second order. 

 To find a consider equation (11), it becomes 



l + p, 4>2-</>i + o-, 20 1 -20 2 + « + T 



20 1 + « 1 +0 1 2 , </>, + 20 2 ^.</> 1 , l-20 1 2 -20, ai -^ 2 -^ 

 The terms of the second order are 



= 0. 



2 2 — 20j -f- a<f> j + 2 0j + 02 — 0l(20[ + «] — 2 ) — O" 



+ (20 1 -20 2 + a)0 1 = O: 

 .'. 0"= (02 — 00(20! + a). 



Hence we see that the scale-reading is (0 2 — X ) (1 4- 20 l +a) 

 and that the horizontal cross-hair of the eyepiece will move 

 across the image of the scale through an angle Sz or 

 8(20, — 2 + a). This remains constant, so that the movement 

 across the scale due to twist of the specimen is very small, 



