igi8 PHILOSOPHICAL TRANSACTIONS. [aNNO 1787, 



rule for finding impossible roots from an equation, whose roots are the squares 

 of the differences of the roots of the given equation ; this rule (as has been ob- 

 served by me in the Miscell. Analyt. and Philos. Trans.) always discovers whether 

 all the roots pf the given equation are possible or not ; and the last term of the 

 resulting equation discovers also, whether 0, 4, 8, 12, &c. or 2, 6, 10, 14, &c. 

 impossible roots, are contained in the given equation ; to which may be subjoined, 

 if the given equation has r possible and n — r = 2i impossible roots, that the 

 number of changes of signs from + to — and — to -j- in the resulting equa- 



tion will not be less than r . — — - , and the number of continued progresses 



from 4- to + 3"d — to — will not be less than t : whence, if the number of 

 continued progresses be t% the number of impossible roots will not be greater 

 than 2t\ and the number of possible roots not less than n — It". If the number 

 of changes of signs be h', the number of possible roots will not be greater than 



/, where / X ~ is the greatest possible number which does not exceed h\ 



and the number of impossible roots not less than n — /. Another rule was, I 

 believe, first given by me in the Miscell. Analyt. 17^2, for finding impossible 

 roots, by finding an equation whose roots are z, where x" — px"-^ -j- q^'"~'^ — 

 &c. = z, and nx"-'^ — (n — 1) px"-^ -j. (7^ — 2) qx"-^ — &c. = 0. 



In the Medit. Algebr. somewhat has been added concerning impossible, 

 affirmative, and negative values of the unknown quantities, in an equation which 

 involves 2 or more unknown quantities ; and also was first delivered a rule from 

 the number of affirmative, negative, and impossible roots of an equation being 

 known, to find the number of impossible, negative, and affirmative roots of an 

 equation, whose roots have a given algebraical relation to the roots of a given 

 equation ; on which two last subjects little, I believe, had been before pub- 

 lished. 



XP^. Experiments on the Production of Dephlogisticated u4ir from Water with 

 various Substances. By Sir B. Thompson, Knt., F. R. S. p. 84. Dated 

 Munich, Sept. 1, 1786. 



Various opinions having been entertained with respect to the origin of the 

 dephlogisticated air, produced by exposing healthy vegetables in water to the 

 action of the sun's rays, according to the method of Dr. Ingenhousz ; and not 

 being myself thoroughly satisfied with any of the theories proposed, I made the 

 following experiments, with a view of throwing some new light on that subject. 



Having found that raw silk possesses a power of attracting and separating air 

 from water in great abundance, when exposed in it to the action of light, it oc- 

 curred to me to examine the properties of this air, and to consider more attentive- 

 ly the circumstances attending its production. 



