DETERMINATION OP THE AMOUNT OF ROTATION 



Let us take the example of the former paper, namely, 9 



<P,= 10.3°; then^,/2 = 7.5° = 7 

 log sin <pJ2 8.9531 

 log cot s^i/2 0.8806 

 log tan ^ 9.8337 



8 = 34° 17' 



30'; <pJ2 = 5.15° = 5' 

 log tan <f J 2 8.9549 

 log sin S 9.7508 



log tan <P 12 9.2041 

 <pl2= 9° 5' 

 cp =. 18° 10'. 



739 



= 15°; 



09'. 



By the simpler, but less accurate, method we find tan ^ = 9^1 9i 

 = 5.15/7.5 = 0.6866; hence ^ = 34° 28'; and 9 = V <pi ^ 9^ 

 = 18.16°, or 18° 10'; the value of <p is the same and that of <5 very 

 nearly the same as those obtained by the accurate method, but the 

 azimuth of the axis, which was taken in the plane of the horizontal and 

 vertical axes, was wrong by ^'2/2, or about 5 degrees. 



Rotations in opposite directions introduce no complications. We can 

 always take a position so that the horizontal axis shall point to the left ; 

 that is, that the forward part of the stratum shall be rotated upward. 

 Then the vertical axis of rotation may either point downward, as in 

 figure 1, or upward, as in figure 2; these two include all possible cases. 

 We use the same formulae for either case, always regarding <Pi and (p^ as 

 positive, the one difference in the results being that the axis OjRwill lie 

 behind or in front of OA, according as the vertical axis points down- 

 ward (figure 1) or upward (figure 2). The angle between Oi^and OA 

 will be ^2/2 in both cases. 



The following table shows values of ^ and <P when calculated by the 

 accurate and by the approximate method (the latter indicated by the 

 primes). The values selected for <Pi and <p2, with the exception of the 

 last, make them together equal to 90° ; if their sums were smaller, the 

 errors would be less : 



^1 



*P2 



<P 



<P' 



Dif. 



d 



d' 



Dif. 



80° 



10° 



80° 30^ 



80° 37^ 



07^ 



5° 56^ 



7° 08' 



1° 12^ 



65 



25 



69 10 



69 38 



28 



18 45 



21 02 



2 17 



50 



40 



63 14 



64 02 



48 



36 16 



38 40 



2 24 



45 



45 



61 38 



63 37 



1° 59 



42 44 



45 00 



2 16 



10 



80 



80 30 



80 37 



07 



82 15 



82 52 



37 



80 



80 



108 08 



113 07 



4 59 



37 27 



45 00 



7 33 



