om - Dr. WARING on the, 
ir. Mr. Evver or Mr. pE 1A Grance found, that if @ bea 
root of the equation x"— 1=0, where # is 4 prime number, © 
a, 27, a5 +. a'—', 1 will be (7) roots of it. More gaya aime 
lar fabject has been added in the laft edition of the Medit. 
Alsebr..) 12.1 Tt is obferved in the Mifcell. Analyt. that 
‘Carpan’sor Scipio FERREUvS’s refolution of a cubic 1s a refo- 
lution of three different cubic equations; and in the Medit. 
Algeb. 1770, the three cubics are given, and the rationale of 
the refolution (for example: Wa, G, and y, be the roots orm 
the cubic equation «+gx—r=o, then is given the function 
of the above roots, which are the roots of the reducing equa- 
tion 8° —rz°=q°); and alfo the rationale of the common refo- 
lution of biquadratics. 13. It is aflerted in the Mifcell. that 
if the terms (My"+ dy"7v +:cy"*x" + &c. and Ny”+ By*—ty 4 | 
Cy”—2x° + &c.) of two equations of ” and m dimenfions, which 
contain the greateft dimenfions of * and y have a common di- 
vifor, the equation whofe root 1s « or y, will not afcend to 
a xm dimenfions; and if the equation, whofe root is x or y, 
afcends to mx m dimenfions, the fum of its reots depends on 
the terms of m and ~—1 dimenfions in the one, and m and 
m—1 dimenfions in the other equation, &c. It is alfo afferted, 
in the Mifcell. that if three algebraical equations of », m, 
and y dimentions contain three unknown quantities «, y, and . 
z, the equation, whofe root is x ory or %, cannot aftend to 
more than 2.m.rdimenfions. 14. Mr. Bezour has given 
two very elegant propofitions for findmg the dimenfions of the 
equation whofe root is x or y, &c ; where x, y, &c. are une 
known quantities contained in two or more (4) algebraical 
equations of =, e, c, &c. dimenfions, and in which fome of 
the unknown quantities do not afcend to the above z, p, o, &c. 
dimenfions 

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