855 



which factors, F is entirely arbitrary. On account of this unre- 

 stricted character of F, innumerable changes may be effected on a 

 pair of conjugate factors, without disturbing their product: but it 

 is from the following expression of these factors that the general 

 results in this paper are chiefly deduced : namely, 



X={F+/+V' P+2F/+/^-X } 1 ,2) 

 X {F+/- ^P + 2I-y+/^-X} J 

 from which it follows that, having decomposed any function X into 

 a pair of conjugate factors (1.), we may always afterwards add any 

 quantity,/, to the rational part of each factor ; provided we, at the 

 same time, introduce the expression 2Ff+/- under the radical : and 

 it is upon this general truth that the results in the paper entirely 

 depend, and from which the rejective intervals are discovered. 



The value of the principle is illustrated by examples taken from 

 Sturm, Fourier, and others ; and some general theorems are de- 

 duced — applicable to all equations — in reference to the existence 

 of imaginary roots, which furnish some remarkably simple criteria. 

 For instance: it is shown that if, in the general equation of the 

 fourth degree, 



3(^-\-ax^^-\-bx''- + cx-\-d=0, 

 in which d is positive, the condition 



^(b—\a^)d>c'^ 



exists, all the roots are necessarily imaginary. And that if, in the 

 general equation of the sixth degree, 



-\- ax'" -\- hx^ + cx^ + dx- + ex +/= 0, 

 in which /is positive, the condition 



^{d{b-la^)-^'^-]f>^{b-\a^-y- 



exists, each member being positive, all the roots are necessarily 

 imaginary. 



The paper concludes with some general propositions, derivable 

 from the principles established in the preceding investigations, 

 and which the author conceives to be of value in the analysis of 

 equations. 



14. " On some Phenomena and Motions of Metals under the in- 

 fluence of Magnetic Force." By William Sykes Ward, Esq. Com- 

 municated by William West, Esq., F.R.S. 



In the course of some experiments relative to the principal phe- 

 nomena of dia-magnetism, the author observed that the nature or 

 direction of the action upon many metals varied with the intensity 

 of the magnetic force, the effects being in accordance with the ob- 

 servations of Professor Plucker ; and in pursuing his researches with 

 the view to ascertain how far the magnetic and dia-magnetic forces 

 might be coexistent in the same metal, other phenomena dependent 

 on the power of the magnet presented themselves. 



On submitting gold, silver, lead, tin, zinc and cadmium to the 

 action of the electro-magnet when excited by an electric current of 

 moderate strength, or when the polar terminations of the magnet 



