0?i the general Solution of Algebraical Equations. 1 65 



pletely developing tlie mutual relations of the current before 

 and during the working of the machine. 



I take the liberty of sending you some memoirs from the 

 Bullctifi scientifiqiie of the Academy. The result of the joint 

 memoir of myself and M. Lenz is that the attraction of electro- 

 magnets is as the square of the force of the current, or as the 

 square of the. electrolytic action of the battery. It appears that 

 this important law holds good for machines in motion ; at 

 least the experiments I have made on that point do not de- 

 part from it more than may be admitted as the error of ob- 

 servation or the result of accidental circumstances. 



I am, &c. 

 St. Petersburg, June 21, 1839. M. H. JacOBI. 



XXX. On the general Solution of Algebraical Tuquations. By 

 J. W. Lubbock, Esq., F.B.S.^ 



T ET X- + Cx^ + D .r + E = (1.) 



-'-' be* any equation, and let 



X = p •{■ qy + ry"- + sy^ +&c.. (2.) 



y being raised to the w— 1^'' power in the last term of equa- 

 tion (2.). Let/j/ = (3.), and let «, /3, y be the roots of 

 equation (3.). Moreover, let 



^1 = a + /3 + y ... &c. S2 = a^ + /3'' + yH &c. 



f ==z a + b + c ...Slc. f - a" + 6^ + c^ + &c. 



a, b, c, &c. being the roots of equation (1.). 



The equation which arises from the elimination of y be- 

 tween (2.) and (3.) will be, as is well known, that formed by 

 the product {Francceiir, vol. ii. p. 12G.) 



[x '— p — qa. — r u^ + &c.} 



with {x - p — q^ — r^'^^ + &c.} 



and all similar quantities ; and as this equation must be iden- 

 tical with equation (1.), 



{x—p — qa — rc^... + &c.} 



{x-p-q^-r^~... + &c.} = A'«... + Ca;- + Da; + E= 0. 



Dividing both sides of the last equation by a;", taking the lo- 

 garithms and equating the coefficients of the same powers of 

 X, we obtain at once the equations of condition which exist 

 between the quantities j), q, r, &c., 2q, Sj, '%q, &c., and the 

 coefficients C, D, E, &c. of the proposed equation. 

 Now let equation 3. be j/" — 1 5= 0, then 



Xo = «, 2i = 0, ^2 = 0, &c. 



* Communicated by the Author. 



