60'2 PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



same book is given a method of muling the aggregates of any algebraical functions 

 of each of the roots of given equations, which is somewhat improved in the lat- 

 ter editions. 



8. In the same book are assumed ^qr^ U, + &c . and ^ + ^_ t ^ & —, 

 where z is any rational quantity whatever, for x and y, the unknown quantities 

 of a given equation of two or more dimensions. 



Q. In the Miscell. Analyt. a biquadratic, a 4 + 'Ipx 3 = qx 2 -\- rx -f- s, of which 

 no term is destroyed, is reduced to a quadratic, x' 1 -\-px -\- n = Vf + c« -f- qx -f- 

 y/r+1? ; and in the 2d edition of it, printed in the years 1767, 1768, 1769, 

 and published in the beginning of the year 1770, the values of n are found 

 _ I— , — — , and — - — ; and the values ot s/ f + 211 + q respectively 



« + /3-y — J" * + y - /3 — 3 a. + $ — /3 — y , , . 



_Z £ f ^ --— , , and their negatives; and the six 



. a.3 — yj 1 «y — /3? u$ — /3y . , . 



values of \/« + z ( 2 respectively — - — , — - — , — - — , and their negatives. 



10. From a given biquadratic, y 4 + qy- -j- ry + s = O, by assuming y- -\- ay 

 + b = v and o and Z> such quantities as to make the 2d and 4th terms of the 

 resulting equations to vanish, there results an equation, v 4 + av" + b = O, of 

 the formula of a quadratic. Mr. De La Grange has ascribed this resolution to 

 Mr. Tschirnhausen ; but in the Leipsic Acts the resolution of a cubic is given 

 by Mr. Tschirnhausen, but not of a biquadratic : his general design seems to be 

 the extermination of all the terms. 



1 1 . Mr. Euler or Mr. De La Grange found, that if « be a root of the equa- 

 tion a" — 1 = 0, where n is a prime number, then a., a. z , a 3 , ..«■-', l will 

 be (n) roots of it. More on a similar subject has been added in the last edition 

 of the Medit. Algebr. 



12. It is observed in the Miscell. Analyt. that Cardan's or Scipio Ferreus's re- 

 solution of a cubic is a resolution of 3 different cubic equations ; and in the 

 Medit. Algeb. 1770, the 3 cubics are given, and the rationale of the resolution 

 (for example : if a, (3, and y, be the roots of the cubic equation x 3 -j- qv — r = O, 

 then is given the function of the above roots, which are the roots of the reducing 

 equation z° — rz 3 = q 3 ) ; and also the rationale of the common resolution of 

 biquadratics. 



13. It is asserted in the Miscell. that if the terms (My" -J- by"-' x -f- cy"~ z x' 1 

 + &c. and Ny m + By" 1 -' x -{- cy m -' l x L -f- &c.) of two equations of n and m di- 

 mensions, which contain the greatest dimensions of x and y, have a common 

 divisor, the equation whose root is x or y, will not ascend to n X m dimensions ; 

 and if the equation, whose root is xory, ascends to n X m dimensions, the 

 sum of its roots depends on the terms of n and n — 1 dimensions in the one, 

 and m and m — l dimensions in the other equation, &c. It is also asserted in 



