i6 L. B. Tuckcnnan 



( A~ , S k +1 )=2 cos b cosr sin- ^ r — cos# sin r 



(6^., A" t + 1 )=2cos£ cos r sin 2 ^r- cos a sinr 



( kS a ., Q k +1 )=2 cos r cos a sin 2 *4 r — cos b sin r ( 29 ) 



(Q k , S K + l )=2cosc cosasin 2 ^r-f cos<$ sinr 



(Q k ,K k+1 )=2 cos, a cosb sin 2 }4r — cosr sinr 



{K k ,Q k _ x )=2 cos a cos b sin 2 : j r — cos c sinr 



Where cos a, cos & and cos c are the direction cosines of the axis 

 of rotation, and r, the angle of rotation. 

 From these, 



cos«=^^^±4^^i^±i> 



smr 



cp „ J:=; / (g^H.) r (^g t+1 ) (30) 



sinr 



co.^^a^r'g"*'-' 



sinr 



Substituting in the above the values of the coefficients in the sec- 

 ond form of the inductive theorem, since that form gives the 

 simpler results, 



sin r=sin2 7r7V r i .. rl 



cos<7=cos2(X\£ — 11 (31) 



cos £=sin 2 ( £,/(■- 1 1 



cos c=o 



If the point S=P, Q=K=o, be chosen as the pole of the sphere, 

 the plane S=o will be the equatorial plane. Letting / represent 

 the latitude and m the longitude of a point on the sphere, then : 



• r S 2<? K . f, 



Sm/ =P=1^? ^=Q=tg20 (32) 



or, <?=tgij'/ m=20 



172 



