12 MeadowcROI'T, Trcatjiient of Geodesic 'J'orsion. 



Then Ox is the generator for which X=0. For G' A' is 

 very small. 



The equation of O'G' is 



X >' - 2Zfc'X' _ z 



I 2^/\' — 2/zX' 



neglecting X'. 



.•. its direction-cosines are a I, 2^A', and —2hX. 

 Those of G are i, O, O. 



.• . the direction cosines of 00' are 



M 



X 



O - 2/l\' - 2i,'X' 



But A : ^t : 7 : : (7 : I : (17 and . •. ,^' = 0. 

 . • . the equation of the hyperboloid is 



by + CZ' + 2_y^'G + 2//.V)' + 2 7f'S = O. 



The tangent plane at (;-, (9, (9) is 



(.V - r)0^y . 2/;/-+ ■; . 2iv= O, 

 or 



jl' . hr + s?£' = O. 



Similarly the tangent plane at (/, O, O) is 



V . hr + s . w= O. 



If these planes are inclined at an angle S we have 



hr . hr + w^ 



J/i^r- + 'cv' J h'r' + 7V' 

 squaring we find 



cos S. 



//W' + 2tv'h'rr' + zv^ = cos"(i(//^;-v-'" + 7U' . Ji'r^ + fvr"^ + w^) 

 . ■ . sin^o(/r/y + 7v-f = cos'?(a' - rfiirJi^ 



. • . (''''' + W ) tan I -\-{r' - r)j = O (x) 



