90 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



ist, If 256**— 1289V + (I44r*9+ 169') X J — 27r* — 4rY be a 

 negative quantity ; then the equation has two, and not more impossible roots. 



2d, If that quantity be affirmative, and either — q or 9' — As be a negative quan- 

 tity; then the given equation will have 4 impossible roots. 



3d, If it be equal to nothing, and either — 9 or 9^ — 4* be a negative quan- 

 tity ; then the two unequal roots of the given equation will be impossible. 



Prob. II. To find how many impossible roots are in the given equation or" -^- 

 qo^ — rx^ -{■ sx — t = 0. 



1st. If the signs of the terms of the equation m;*° -f lOqw"^ -f- {^Qq"^ -\- \0s) X 

 w^ -f (8O9'' + 50qs -\- lor'') X vP -\- {Q5q* + ilAq's — QSi' -|- 929/-^ + 200rt) 

 X m;" + (669* — 36O9A* 4- 1969^* + 1 189^7-^ -f 260r-s + 625^' -I- AOOqrt) X 

 «;* -f- (259" -I- 40/ — 53r* + 52q^r^ — 5229V -f- ig4q*s + JOSqAi -\- 240q^rt 

 + \75Oqe — g50*r/) X w" + {Aq' -\- lOQq^s — 80q/ — SOSq^s' — I02qr* — 

 7q*r'' -)- 570rV -|- 6l2q''r''s + yoOr'^t — 3750i'i -|- 2500<Y + SOr^ — 2150 

 qrst) X U!^ -\- (4005 — SOOqV — 159V -j- 249"* —Sq^r" — Abq'^r* — 270r*s -f 

 140/^Y + 960rV9 -f 1875«V^ + lOOO^ri' — 5000^Y -f- 1750^^9' -f 40/r9^ -j- 

 600<r^9 — \650trsq'') -|- w^' -f- (369V — 2249V -j- 32O94* -f 49^" -|- 27r<' — 

 40rV -t- 434rY'5'^ — 24r^59^ — 198r''9i -j- 5000fV — 450ir^J — 6250/V -f- 

 675<^9'* — STbOt^s + 3000<V^9 -f 60<r^9" -\- 200trs^q — 330trq^s) X w -f- 

 3125^* — 3750qrt^ -f- (2000**9 -\- 2250r's — QOOsq^ -f 825rY + IO89') X t' 



— (1600/r — 560r9^*' — l6r^9^ -|- 630r^9* +■ 72r59* — 108r«) X t + 256** 



— 1289'i*-f- 1447-^9/ -f- 169"*' — 27rV — 4r'9V = 0, be continually changed 

 from -f- to — , and from — to -|- ; the given equation has no impossible roots. 



2d. If the signs of the terms of the equation be not continually changed from 

 -j- to — and — to -f ; then 2 or 4 of the roots of the given equation will be 

 impossible, according as the last term of it is negative or affirmative. 



of the Phil. Trans., and others he published in separate works : as, the Meditationes Algebraicae, in 

 1770; the Proprietates Algebra icarum Curvarura, in 1772; and the Meditationes Analytic^, in 

 1776'. To these might be added a work written in his retirement, on morals and metaphysics; of 

 which a few copies only were printed, and presented to his friends. For his various ingenious pa- 

 pers in the Phil. Trans., Dr. W. was, in 1784, deservedly honoured by the r. s. with their gold 

 medal. And most of these essays give strong proofs of the powers of his mind, both in abstract 

 science, and its application to philosophy : though they labour, in common with his other works, 

 under tlie disadvantage of being conveyed in a very unattractive form. As to the contents of the 

 separate volumes above mentioned, a very full analysis of them may be seen in Dr. Hutton's Dic- 

 tionar}', vol. 2, p. 717, as communicated by Dr. W. himself In his disposition and character Dr. 

 W- is represented as of inflexible integrity, great modesty, plainness, and simplicity of manners; of 

 a meekness and diffidence of manner to such a degree, as to be always embarrassed before strangers. 

 His extreme short-sightedness too, joined to the natural want of order and method in his mind, 

 which appeared remarkably even in his hand-writing, rendered his mathematical compositions so 

 confused and embarrassed, that in manuscript they were often utterly inexplicable : a circumstance 

 wjiich may account for the numerom typographical errors in hit publications. 



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