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On the geometrical Representation of the Powers of Quant itir.t, u'l.ose 

 Indices involve the Square Roots of negative Quantities. By the 

 Rev. John Warren, M.A. late Fellow and Tutor of Jesus College, 

 Cambridge. Communicated by the President. Read June 4, 1829. 

 [Phil. Trans. 1829,^. 339.] 



The author, in a former paper, read to the Society in February last, 

 had discussed various objections which had been raised against his 

 mode of geometric representation of the square roots of negative 

 quantities. At that time he had only discovered geometrical repre- 

 sentations for quantities of the form a + b \/ 1, of geometrically 

 adding and multiplying such quantities, and also of raising them to 

 powers either whole or fractional, positive or negative ; but he was 

 at that time unable to represent geometrically quantities raised to 

 powers, whose indices involve the square roots of negative quantities 



. m + n ^/ 1 



(such as a + i v 1 ) His attention has since been 



drawn to this latter class of quantities by a passage in M. Mourey's 

 work on this subject, which implied that that gentleman was in posses- 

 sion of methods of representing them geometrically, but that he was at 

 present precluded by circumstances from publishing his discoveries. 

 The author was therefore induced to pursue his own investigations, 

 and arrived at the general result stated by M. Mourey, that all alge- 

 braic quantities whatsoever are capable of geometrical representation 

 by lines all situated in the same plane. The object of the present 

 paper is to extend the geometrical representations stated in his 

 former treatise, to the powers of quantities, whose indices involve the 

 square roots of negative quantities. With this view he investigates 

 various equivalent formulae suited to the particular cases, and employs 

 a peculiar notation adapted to this express purpose ; but the nature of 

 these investigations is such as renders them incapable of abridgement. 



An experimental Examination of the Electric and Chemical Theories 

 of Galvanism. By William Ritchie, A.M. F.R.S. Rector of the 

 Royal Academy at Tain. Read May 7, 1829. [Phil. Trans. 1829, 

 p. 361.] 



After observing that the theory of galvanism originally proposed 

 by Volta, and generally termed the Electric theory, is still the uni- 

 versally received doctrine among continental philosophers, the author 

 adduces several experiments proving the fallacy of the principles on 

 which that theory is founded. He points out the inconclusiveness 

 of the reasoning by which it has been inferred that dissimilar metals, 

 by being simply placed in contact with one another, are instantly 

 thrown into opposite electric states ; for in all the experiments which 

 have been made with a view of establishing this fundamental prin- 

 ciple of the electric theory, the metals have been exposed to the oxi- 

 dizing action of the air, which is a constant source of electric disturb- 

 ance, and the operation of which has been strangely overlooked. The 



