VOL. LXIX.J PHILOSOPHICAL TRANSACTIONS. 487 



represent the motion of the comet at different periods of time during its appear- 

 ance, that its orbit may be sensibly elliptical, which it seems M. Pingre, who 

 first calculated the orbit in a parabola, had also some suspicion of, and concludes 

 with recommending the investigation of the true elements of its orbit in an 

 ellipsis. The laborious calculation thus recommended has, we see, been since 

 successfully and satisfactorily performed in this paper by Mr. Lexell. 



N. Maskelyne. 



IX. On the General Resolution of Algebraic Equations. By Edw. Waring, 



M.D., F.R.S., &c. p. 86. 



In the year 1757 I sent some papers to the r. s,, which papers were printed in 

 the year 1759, and copies of them delivered to several persons; these papers 

 somewhat corrected, with the addition of a 2d part, on the properties of curve 

 lines, were published in the year 1762. In the years 1767, 1768, and I769, I 

 printed, and published in the beginning of the year 1770, the same papers with 

 additions and emendations, under the title of Meditationes Algebraicae. In these 

 papers were contained, with many other inventions, the most general resolution 

 of algebraic equations known, as it contains the resolution of every algebraic 

 equation, of which the general resolution has been given, viz. the resolution of 

 quadratic, cubic and biquadratic, the resolution of Mr. de Moivre's and Mr. 

 Bezout's, since published, equations; it discovers the resolution of an equation 

 of n dimensions, of which the n roots are given, and also deduces innumerable 

 equations of n dimensions, which contain n — 1 independent coefficients. 

 Whence it seems probable, that this new method of mine may contain the most 

 general resolution of algebraic equations that ever has, or perhaps ever will be 

 invented. 



The general resolution \s x = a ^p + i>l/p'' + c ^p^ + d l/p* . . . . + 

 r^p'—i -f s^/p"-"- + tl/p"-' -f -, if the equation be x" — ax"-' + bx"-^ — 



cx"-i + HX"'^ — &c. = o. 



I shall add the resolution of some particular equations from this method, and 

 then subjoin the equation to which x = a^p + b'^p'^ + '^\^P^ + &c. is the 

 general resolution. 



1 . Let the resolution be a? =: a4^p + b^p^'. then the correspondent equation 

 free from radicals will be found x^ — 3abpx — a^p — b^p"^ = o. Let x^ — vx — 

 Q = o be a cubic equation whose resolution is required, which suppose the same 

 as the equation found above, and consequently their correspondent terms equal, 

 i. e. p = 3abp, and a = a^p + b^p-; whence/) = —7, which value being substi- 



tuted (orp in the 2d equation, there results a = -7- + 3-^. In this equation for 

 a or b may be assumed unity, or any other quantity whatever, and there will 



