Rigid Body about a Fixed Point. 443 



" The arcs joining the extremities (each with each in either 

 order) of two other equal arcs, subtend equal angles at either 

 of the points of intersection of two great circles bisecting at 

 right angles the first-named connecting arcs*." 



The spherico-triangular mode of compounding rotations 

 given in the above simple disquisition may easily be made the 

 parent of a whole brood of geometrical consequences, which, 

 however, I must leave to the ingenuity and care of those who 

 have a turn for this kind of invention. 



But I ought not to omit to invite attention to a remarkable 

 form, which may be imparted to the theorems above stated 

 for the composition of finite rotations, or rather to a theorem 

 which may be derived from them by an obvious process of 

 inference. 



Let P, Q, R . . . XZ be any number of points on a sphere 

 capable of moving about its centre, joined together by arcs of 

 great circles so as to form a spherical polygon. Imagine any 

 number of rotations to take place about these points in suc- 

 cession as poles. It matters not which is considered the first 

 pole of rotation, but the order of the circulation must be sup- 

 posed given, as, for instance, P, Q, R . . . XZ, or QR...XZP, 

 or R...XZPQ, &c« This will be one order ; the reverse order 

 would be PXZ ... RQ, or QPXZ ... R, &c. 



I shall suppose the circulation to be of the kind first above 

 written. Now we may make two hypotheses: — 



1. That the poles are fixed in space. 



2. That they are fixed in the rotating body. 



In the first case, let the rotations about the given poles P, Q, 

 R, S ... XZ be double the amounts which would serve to 

 transport PQ to QR, QR to RS . . . XZ to ZP respectively. 



In the second case, let the rotations be double the amounts 

 which would carry PZ to ZX.. . . SR to RQ, RQ to QP 

 respectively. Then, on either supposition, the sum of the 

 combined rotations is zero ; or, to use a more convenient and 

 suggestive form of expression, if the poles of rotation form a 

 closed spherical polygon whose angles are respectively equal 

 to the semirotations about the poles, the resultant rotation is 

 zero. 



This proposition is immediately derivable from the funda- 

 mental one relative to three poles, given above^ by dividing 

 the polygon into triangles by arcs, joining any one of the poles 



* This proposition will be seen to be immediately demonstrable, by the 

 comparison of equal triangles, when viewed as the converse of this other. 

 *' The arcs (or right lines) joining the correspondent extremities of the bases 

 of two similar isosceles spherical (or plane) triangles having a common 

 vertex, are equal to each other." 



