VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 603 



the Miscell. that if 3 algebraical equations of n, m, and r dimensions, contain 3 

 unknown quantities x, y, and z, the equation, whose root is x ory or z, cannot 

 ascend to more than n . m . r dimensions. 



14. Mr. Bezout has given two very elegant propositions for finding the di- 

 mensions of the equation whose root is x or y, &c. ; where x, y, he. are un- 

 known quantities contained in two or more (h) algebraical equations of ir, f, a-, 

 &c. dimensions, and in which some of the unknown quantities do not ascend to 

 above the w, g , <r, &c. dimensions respectively. In demonstrating these proposi- 

 tions he uses one (among others) before given by me (viz. if an equation of n 



dimensions contains m unknown quantities, the number of different terms which 



i .• i • -i -Hi / • \ »+2 ra + 3 n 4- m. T , «»■• 



maybe contained in it will be (n + 1) . — - — . — - — . . — — ). In the Medit. 



1770 there is given a method of finding in many cases the dimensions of the 

 equation whose root is x or y, &c. ; from which one, if not both, of the above- 

 mentioned cases may more easily be deduced, and others added. 



15. In the Medit. 1770 it is observed, that if there be n equations containing 

 m unknown quantities, where n is greater than m, there will be n — m equations 

 of conditions, &c. — 16. In the Miscell. is given and demonstrated the subse- 

 quent proposition ; viz. if two equations contain two unknown quantities x and y, 

 in which x and y are similarly involved ; the equation whose root is x or y will 

 have twice the number of roots which the equation, whose root is x + y, x'~ + 

 w 2 , &c. has. In the Medit. 1770 the same reasoning is applied to equations, 

 which have 2, 3, 4, &c. quantities similarly involved. 



17. Mr. De La Grange has done me the honour to demonstrate my method 

 of finding the number of affirmative and negative roots contained in a biquadratic 

 equation. A demonstration of my rule for finding the number of affirmative, 

 negative, and impossible roots contained in the equation x" -f xx m + b = O is 

 also omitted, on account of its ease and length. From the Medit. the investi- 

 gation of finding the true number of affirmative and negative roots appears to be 

 as difficult a problem as the finding the true number of impossible roots ; and 

 it further appears, that the common methods in both cases can seldom be de- 

 pended on. But their faults lie on different sides, the one generally finds too 

 many, the other too few. 



18. In the Medit. 1770, from the number of impossible roots in a given equa- 

 tion fr" px"~ l + &c. = 0) is found the number of impossible roots in an 



equation, whose roots (v) have any assignable relation to the roots of a given 

 equation ; and examples are given in the relation (nx"- 1 — (n — l)px'-x + &c. 

 = v) ; and in an equation, whose roots are the squares of the differences of the 

 roots of the given equation. — 19. It is observed in the Medit. 1770, that in 

 two or more equations, having two or more unknown quantities, the same 



4 h 2 



