4,gf4 PHILOSOPHICAL TIIANSACTIONS. [aNNO 1779, 



which call respectively c, d, e, &c.: then will the equation required be x" ;fi — 

 np (au + bv + ct + ds + &c.) x"-"- — xx'-i + BX'-'t — cx-'S + d^'-^- — 

 &c. = O. 



From the same principles may be deduced the most general reduction yet 

 known of equations to others of inferior dimensims, e. g. Let (x) ^'' + (a -f. 

 niyp + b^l/p" + c^/)^ + . . . + s';/li"-^ + tX/p"-') x"'' + (B + fl>/) + 

 b"l/p' + . . . + /"^//'-^ + t""^p-^) x"-' + (c + u""^p + b-l^p' + &c.) 

 x'-i + &c. = 0; let a, [3, y, iJ, &c. be the respective roots of the equation z" — 

 1=0; then, from the principles before given, may be formed the different 

 values of the equation x, which being multiplied into each other, from the pro- 

 positions before-mentioned of the Meditationes Algebraicae, may be deduced an 

 equation of mn dimensions free from radicals, whose root is x, and which con- 

 tains nm unknown quantities a, a, b, c. Sec. b, «', b', c', &c. c, a", 6", c", &p: 

 for one, two or more of these unknown quantities may be assumed any quan- 

 tities whatever, and thence may be deduced equations of Jim dimensions, which 

 may be reduced to equations a." + (a -|- a"^ f) -f- b"^/// -f c"^// + &c.) jc"-' -f- 

 &c. = O of 7i dimensions. 



In the same manner may be assumed equations, which involve 1/p, 1/p^, . . ., 



V/)"-'; ;/«, ;/a% [/»', ■■■ ;/a^-'; ^r, ^r\ ^t\ ^r'", &c.; and 



from so reducing them as to exterminate the irrational quantities, may often be 

 derived equations whose resolutions or reductions are known. 



The method of transforming algebraical equations into others, whose roots 

 bear any assignable algebraical (but not exponential) relation to the roots of a 

 given algebraical equation, first published by me in the papers sent to the r. s., 

 and afterwards in the year 1760; and 3dly in my Miscellanea Analytica; and 

 lastly in the Meditationes Algebraicae, and since published by Mr. Le Grange in 

 the Berlin Acts, is perhaps, as Mr. Le Grange obsenes, more general than Mr. 

 Hudde's, or any transformation yet invented; it is very useful in the resolution 

 of numerous problems; and further has this peculiar advantage over all other 

 transformations yet invented, that it often easily discovers some of the first terms 

 of the equation required, from which many elegant theorems may be derived." 



In the works above-mentioned, viz. Miscell. Analyt. Medit. Algeb. &c. are 

 given some problems serving to this transformation ; the first of which is a series 

 which, from the coefficients of a given algebraical equation {x" — px"-^ -{- 

 qx"~- — &c. = O) finds the sum of any power of the roots (viz. a"' + P™ -f- ■j/" 

 -J- ^ -|- &c. where a, p, y, <^, &c. denote the roots of the given equation), the 

 law of which series was published by me many years before it was given by Mr. 

 Euler. The 3d problem, often mentioned in this paper, is an elegant and useful 

 series for finding the sum of quantities of the following kind, viz. x"'p''y'S' &c. 



-I- oc'fi'^y'r &C. 4- a'|3'"y'<J' &C. + a'lS^y'J' &C. + »"i3'y'<5' &C. -f &C. 



