sin a = cos A . tan i 

 sma'= cos A .tan 



Dr. Hirst on Equally Attracting Bodies. 169 



same operations in the triangles AB'C and B'C'D', we arrive at 

 the simple formulae, 



:;.} ■ • • • (^' 



From the centre M of the sphere, let the normals MO, MN to 

 the planes of ABB', ACC be so drawn that the angle OMN 



may be acute, and consequently equal to A. Let us further 

 agree that the acute angles a, a' — and hence also «, a' — shall be 

 estimated as positive or negative magnitudes according as the 

 corresponding arcs fall on the opposite or on the same side of 

 the plane ABB' as does its normal MO. Under these conditions 

 it is evident that the direction angles of the normal MN, with 

 respect to the rectangular axes MO, MB, MB', will be respect- 

 ively equal to 



so that by the equations (3), the direction cosines will have the 

 respective values, 



cos A, cos A . tan a, cos A . tan a'. . . . (4) 



11. Conceive M to be a point on any surface, MO the radius 

 vector making an angle 6 with the polar or a;'-axis, (f> the angle 

 between the plane {xy) and the vector-plane through the a?-axis. 

 Further, let MB and MB' be drawn perpendicular to the radius 

 vector, the first in the above vector-plane, the second perpendi- 

 cular thereto, and both in the direction of the increasing angles 

 6 and (p. If, lastly, we identify the tangent-plane at M with 

 that of our former great circle ACC, it will be at once seen that, 

 iu virtue of the agreement respecting the signs of a and «', 



Idr 

 ^^'''' = rdB' 



and ^ , 1 dr 



tan a' = 



rsindd^' 



or, if we set , r 



H- = u, (5) 



where c is an arbitrary constant or line, 



du 

 tan« = ^-g, 



, I du 



tan ft' = ^-7, --. 



sin a dip 



