408 



some theorems lately obtained by means of the quaternion 

 analysis. 



1. {Rule of Derivation.') — Let CD be a given right line, 

 bisected in L, and LI a given perpendicular thereto. Assume 

 at pleasure any point P in the same plane, and derive from it 

 another point Q, by the conditions that 



CQJ.DP, CQ.DP= CD.LI, 



and that the rotation from the direction of CQ to the direction 

 of DP shall be towards the same hand as that from LD to 

 LI. From Q derive It, and from R derive S, &c, as Q has 

 been derived from P. It is required to investigate geome- 

 trically the chief properties of the restdting arrangement. 



2. (First Case: LI < LD). — If the given line LI be less 

 than LD, then, parallel to the 

 latter line, there can be drawn, 

 through the extremity I of the 

 former, a chord AB of the cir- 

 cle (L), described on CD as 

 diameter : and we may sup- 

 pose that the point B is nearer 

 than A to C. Then, 



CQ.DP= CA.DA = CB.DB, 

 and ACQ = ADP, BCQ = BDP, 



even if signs of angles (or directions of rotation) be attended 

 to. Thus the two triangles ACQ, PDA, and in like manner 

 the two triangles BCQ, PDB, are equiangular, but oppo- 

 sitely turned, like a figure and its reflexion in a plane mirror, 

 or like the two triangles ABC, BAD : which relations we 

 may perhaps not inconveniently express, by saying that in 

 each of these three pairs the two triangles are inversely similar, 

 or by writing, 



ABCk'BAD, ACQk'PDA, BCQk'PDB; (1) 



and then either of these two latter formulae of inverse simila- 



