Motion of a Rigid System about a Fixed Point. 429 



BE = AB and BD=CB. Also produce AC both ways, and 

 make AF=CG= AC. Then because the angles D, E are re- 

 spectively equal to the angles at C and A, and DE =AC, it 

 is plain that the rotation AE (or 2AB) will carry that arc on 

 the moveable sphere which at first coincides with FA, into 

 the position DE; and then the rotation DC (or 2BC) will 

 carry the same arc from the position DE to the position CG. 

 But this last is the position into which it would have been 

 carried by the single rotation FC (or 2 AC) ; whence the 

 truth of the theorem is manifest. 



The following is an equivalent theorem, easily deducible 

 from the preceding. It is worth while, however, to establish 

 it independently. 



Let PQR be any spherical triangle upon the Jixed sphere, 

 the letters being so arranged that Q is the positive pole of a ro- 

 tation from QP to QR. Then, j/ P> Q» R denote the interior 

 angles of the triangle, a positive rotation 2P round the pole P, 

 folloisced by a positive rotation 2Q roimd Q, produces the same 

 displacement as a positive rotation 2(7r — R) round R. 



Demonstratio7i. — p- „ 



Produce PR,and °* 



draw PV making 

 an angle VPQ = 

 QPT. Produce 

 QR; draw Qp 

 = QP,makingan 

 angle ;;QR= 

 PQR; join pR 

 and produce it. 

 Then it is plain 

 that the triangles 

 pQR, PQR have 

 their sides and 

 angles respect- 

 ively equal. Now 

 the first rotation 

 (round P) will carry that arc on the moveable sphere which 

 at first coincides with PT, into the position PV; and the 

 next rotation (round Q) will carry the same arc into the po- 

 sition j9 W. But this same displacement would evidently have 

 been produced by a single rotation round R through the angle 

 PR/;, or 2SRP. Whence the theorem is proved. 



It is easy to establish corresponding propositions for the 

 case in which the spherical triangles are drawn upon the 

 moveable sphere. In fact, we have only to look upon the dis- 

 placements supposed in the preceding demonstrations, as r^/a- 



