XXVIII. On the geometrical representation of the powers of, whose indices involve the square roots of negative quantities

About three months ago I wrote a paper intitled "Consideration of the objections 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 a + b √— 1, and of geometrically adding and multiplying such quantities, and also of raising them to powers, either whole or fractional, positive or negative; but I was not then able to represent geometrically quantities of the form a + b √ — m + n √ — 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 forcé à passer sous silence plusieurs espéces de formules, telles sont celles-ci a √ — 1 , a √ — 1 sin (√ — 1) &c., &c., &c. Je les discute amplement dans mon grand ouvrage, et je démontre que toutes expriment des lignes directives situees sur le mêrae plan que 1 et 1.” where a √ — 1 and 1 1 in M. Mourey’s notation signify respectively a ( 1 1 ) √ — 1/4 and ( 1 1 )according to my notation.