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XXXII. 



AMENDMENT TO " THE TWELFTH AKD COIs^CLUDI^^G 

 MEMOIR Oy THE THEORY OE SCREWS": TRAls^S. 

 R.I.A., Vol. xxxT., pp. 145-196. Et SIR ROBERT BALL, 

 LL.D., F.R.S., Lowndean Professor of Astronomy and Geometry 

 at Cambridge. 



[Eead April 25th, 1898.] 



I HAD occasion to set doTvn, on page 172 of the above-mentioned Paper, 

 a construction for the three double points of two plane homogi-aphic 

 systems. It is no doubt true that the three double points possess the 

 property there stated, but the converse does not generally hold, as 

 I had incautiously supposed. The construction given is therefore 

 unsound. 



A correct construction for the double points of two homographic 

 systems of points in the same plane is as follo"srs : — 



Let and 0' be a pair of corresponding points. Then each ray 

 through will have, as its correspondent, a ray through 0'. The 

 locus of the intersection of these rays will be a conic S. This conic 

 S must pass through the three double points, and also through 

 and 0'. 



Draw the conic S', which is the locus of the points in the second 

 system corresponding to the points on S, regarded as in the first 

 system. Then since lies on S, we must have 0' on S'. But S' 

 must also pass thi-ough the three double points. 0' is one of the four 

 intersections of S and S', and the three others are the sought double 

 points. Thus the double points are constructed. 



On the same page {loc. cit.) I also set down a construction for the 

 four double points of two homographic systems in space. Here again 

 the process is not generally applicable. I replace it by a correct 

 construction, as follows : — 



Let and 0' be two corresponding rays. Then any plane through 

 will have, as its correspondent, a plane through 0'. It is easily seen 

 that these planes intersect on a ray which has for its locus a quadric 



