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XXVIII. On the geometrical representation of the powers of quantities, whose 

 indices involve the square roots of negative quantities. By the Rev. John 

 Warren, M.A. late Fellow and Tutor of Jesus College, Cambridge. Com- 

 municated by the President. 



Read June 4, ] 829. 



About three months ago I wrote a paper intitled " Consideration of the ob- 

 jections raised against the geometrical representation of the square roots of 

 negative quantities," which paper was communicated to the Royal Society by 

 Dr. Young, and read on the 19th of February last. At that time I had only 

 discovered the manner of representing geometrically quantities of the form 

 hr{- b ^ — \, and of geometrically adding and multiplying such quantities, 

 and also of raising them to powers, either whole or fractional, positive or ne- 

 gative ; but I was not then able to represent geometrically quantities of the 



-m +TC ^ — 1 



form a-\-b ^ — \ , that is, quantities raised to powers, whose indices 



involve the square roots of negative quantities. My attention, however, has 

 since been drawn to these latter quantities in consequence of an observation 

 which I met with in M. Mourey's work on this subject (the work which I 

 mentioned in my former paper) ; the observation is as follows : 



" Les limites dans lesquelles je me suis restreint m'ont forc6 a passer sous 

 silence plusieurs espfeces de formules, telles sont celles-ci 



a , a , sin (^ — l) &c., &c., &c. 



V — 1 



Je les discute amplement dans mon grand ouvrage, et je d^montre que toutes 

 expriment des lignes directives situees sur le meme plan que 1 et 1." 



where « ,^rr ^^^ ^ ^^ ^- Mourey's notation signify respectively a {\\ 

 and /ly according to my notation. 



a/-1 



2x2 * ^ 



