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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, 1 829. 
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 
«-{-&>/ — 1, 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 n */ — 1 . .. 
form a + b — 1 , 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 force a passer sous 
silence plusieurs especes de formules, telles sont celles-ci 
V- i 
a , a 
, sin (*V — l) &c., &c., &c. 
Je les discute amplement dans mon grand ouvrage, et je demontre que toutes 
expriment des lignes directives situees sur le meme plan que 1 et 1.” 
where a and 1 in M. Mourey’s notation signify respectively a 
and ^iy according to my notation. 
V- i 
2x2 
