TRANSACTIONS OF SECTION A. 



579 



Department for Mathematics and General Physics. 



1. On Centres of Finite Twist and Stretch. 

 By Professor R. W. Genese, M.A. 



Giren two directly similar figures ABC..., A'B'C... in a plane; it is known 

 that one may he transformed into the other by a twist about a certain point I, and 

 a proper stretch from that point. But it will be seen that the transformation 

 may be effected by an equal twist about any point in the plane (so as to bring 

 AB parallel to A'BO follmoed by a stretch from a different centre S, or equally 

 well by a stretch from S folloxoed by a twist round a proper point 0' not coincident 

 with O. It woidd seem worth while to consider the relations between the 

 centres. 



Fig. 1. 



Fig. 2. 



Let I be brought to J by the twist to round ; then it must be brought back 

 by the stretch from S. Thus it is clear that for a given txcist and stretch the 

 triangle OIS is of constant species. 



Similarly if I be brought by the stretch first to J' it must be brought back by 

 the rotation about 0'. 



It is at once seen that (1) S, 0, 0' are collinear, (2) SO' is the stretched 

 position of SO, (3) 10, 10' are equally inclined to SI and are to each other 

 as SO : SO', so that 0' is the position of O after the first iioist and stretch. 



If we start again with two symmetrically similar figures, one may be trans- 

 formed into the other by a half turn about any axis in the plane, together with a 

 twist and stretch from the proper centre, and 

 the relation between the axis and the centre 

 might be worth investigation. For the pre- 

 sent paper, however, it suffices to observe that 

 by taking an axis parallel to the bisector of 

 the angle between two corresponding sides the 

 twist may be avoided ; it is not difficult to 

 show that the locus of the stretch-centres for 

 axes parallel to this axis is a straight line at 

 right angles to it. In the case in which the 

 stretch-centre is on the half-tuvn axis, the locus 

 and axis form two remarkable straight lines 

 which should bear the name of the illustrious 

 mathematician Bellavitis who first noticed 

 them. 



Their existence may be briefly demonstrated, thus : 



Let AB=jo, A'B'=ju' be corresponding sides, and let AA', BB' be divided at 

 li, M so that AL : IjL' = p : p' =-BM. : MB'. Then LM shall be equally inclined 

 to p,p'. 



p p 2 



