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some theorems lately obtained by means of the quaternion 
analysis. 
1. (Rule of Derivation.)—Let CD be a given right line, 
bisected in Z, and LJ 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 
CQLtDP, 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 R, 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 resulting arrangement. 
2. (First Case: LI < LD).—If the given line LJ be less 
than LD, then, parallel to the 
latter line, there can be drawn, 
through the extremity J of the 
former, a chord AB of the cir- 
cle (LZ), described on CD as 
diameter: and we may sup- 
pose that the point B is nearer 
than 4 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 4CQ, 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 inve/‘sely similar, 
or by writing, 
ABCa’ BAD, ACQa«'PDA, BCQa«'’PDB; (1) 
and then either of these two latter formule of inverse simila- 
