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Mathematics. — "()n a (/ccoui/xtsif/d// of a ciinlutacnis nioft'nn 

 aljuiit II I'l.rril point (f o/' N, ii/fo tiro ront/iiiioiis motions 

 (il)Out O of S./s' by Mr. T.. E. J. 1>rol"wek, comiinuiicated 

 by Prof. KoHTKWKfi. 



(Gominiinicaled in tljc meeling of February 27. lOOi). 



Two planes in S', makinu' tNvo equal angles of position are called 

 mulually 'equiangular to the right" if one is (with its normal plane) 

 |)lane of rotatioji for an equiangular double rotation to the right 

 aboul ihe other one and its normal plane. 



We will call the sides of one and the same acute angle of position 

 "corresponding vectors" through the point of intersection of two 

 equiangular intei'secting planes. 



As is kno\Mi a system of planes equiangular to the right or to 

 the left is intinite of order two. Of course a determined equiangular 

 svslem of [»lanes to the right caji have ^\•itll a determined equian- 

 gular system of |)lanes to the left not more than one i)aii' of planes 

 in common (two paiis of planes cannot intersect eacli other at the 

 same time e(piiangulai-ly to the right and to the left) : but one 

 jiaii' t»f planes they always have in common. We will sliow how 

 tliat comuKui pair of planes can be found. 



A pair (»f intersecting pairs of planes of both systems is of course 

 easy to lind. We lay through any vector (t(' ihe planes belonging 

 to the two systems : their normal jdanes intersect each other in a 

 second vector OJJ. Thus OCD is one plane of position of those 

 two pairs of planes. In the second ]dane of position the four planes 

 furnish four lines of intersection, let us say (JH, <iF, (U\, OG, 

 in such a way, thai the considered pairs of i)lanes must be OCH; 

 Ol)K and ()('F: (>DG. The following figures are supposed to be 

 situated in those two planes of position. 



Fig. 1. 

 Let the pair of planes OCR, ODK belong to the given system 

 equiangular to the riglit, and UCF: OBG to the given system 

 equiangular to the left. 



