428 Prof. Donkin on the Geometrical Laws of the 



monstraiion of it, after establishing the necessary definitions 

 and conventions. 



In considering the motion of a rigid system about a fixed 

 point, let us conceive a sphere to be described about that 

 point as a centre, and fixed in space ; and another sphere, 

 with the same centre and radius, to be invariably connected 

 with and carried by the moveable rigid system ; so that the 

 motion of this latter sphere may be substituted for that of the 

 rigid system, so far as its geometrical displacement only is 

 concerned. For convenience, let the radius of each sphere 

 be taken equal to the linear unit. 



When a rotation takes place about any axis passing 

 through the centre, let the points of intersection of that axis 

 with either sphere be called the poles of the rotation ; and let 

 that be called the positive or no}-th pole, with respect to which 

 the rotation has the same direction as the earth's diurnal 

 motion with respect to its north pole. 



Suppose A, B are any two points on either sphere; we may 

 use the abbreviated expression "the rotation AB" to denote 

 that rotation of the moveable sphere round the axis of the 

 great circle joining AB, which would cause the point A on 

 the moveable sphere to ilescribe the arc AB on the fixed 



sphere ; and we may call the positive or north pole of such a 



rotation the posit irie or north pole of the arc AB. 



It is plain that the rotation AB is the same thing whether 



the two points A, B be supposed to be on the fixed or move- 

 able sphere. But two successive rotations AB, CD will in 



general produce a different displacement, according as the 



four points A, B, C, D are on one or the other sphere. 

 1 now proceed to the theorem in question. 

 THEOREM. 



L,et ABC be any 



spherical triangle 



upon the Jixed 



sphere. Then twice 



the rotation AB, 

 followed by twice 



the rotation BC, 



produces the same 



displaceinent as 



twice the rotation 



AC. 

 Demonstration. 



— Produce AB, 



CB (fig. 1) to E 



and D, making 



