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XIII. Geometrical Interpretation of 'log \Jq. 

 By Alexander Macfarlane. 



To the Editors of the Philosophical Magazine, 



Gentlemen, 



IN the notice of Dr. Molenbroek's Anwendung der Qua- 

 ternionen auf der Geometrie, your reviewer says, " It 

 would probably baffle even a Hamilton to give a geometrical 

 interpretation of logUg" (Phil. Mag. vol. xxxvii. p. 333). 

 As this matter has been treated of in several of my papers, I 

 send you the interpretation required. 



The general quaternion q may be analysed into the pro- 

 duct of a ratio and a versor ; by U</ is meant the versor. 

 Let a denote the axis of the versor and A its amount in 



rr 



radians, then JJq=a A and log Ug = Aa / ; but log« f — ah 



IT 



therefore \og\Jq = Aa~2. A more correct definition of A is 

 the ratio of twice the area of the sector to the square of the 

 initial radius ; for that definition applies also to a hyperbolic 

 versor. 



The geometrical meaning of the above expression will 

 become evident on considering the more general versor given 

 by an equiangular spiral. Let a quinion be denoted by q f y 

 and let it be defined to be such that 



•n 



log U(/ = Aa w =A cos w + Asinw .a*; 



it 



we then find that A sin w . a% is the logarithm of the angle 

 and A cos w the logarithm of the radius of an equiangular 

 spiral of axis a and constant angle w, the initial radius being- 

 unity. Thus w is the constant angle between the radius- 

 vector and the tangent, or rather the difference of the angle 

 from the initial radius to the tangent and that from the 

 initial radius to the radius-vector. In the case of the circle 

 this difference angle is a quadrant: this is the explanation of 

 the quadrantal versor in log U^. In the spiral the quantity 

 A is the magnitude of the complex logarithm, and a w gives 

 the components of the logarithm. The expansion depends on 

 the scalar component of the logarithm, while the rotation 

 depends on the vector component. In the case of the circle, 

 that is of JJq, the scalar logarithm vanishes. 



For further elucidation of this matter consider a hyperbolic 

 quaternion. Let p denote such a quaternion ; when the multi- 



1T 



plier is removed we have Up = a iA and therefore log JJp = iAu?. 



