EQUATION. 



products of every two with their signs 

 changed) = 3m* 3 n, and | = m 1 



-|- n ; also, r (the product of all the roots 

 with their signs changed) = 2 m3 



6 m n, and - = wi3 3 m n ; and by invo- 

 lution, 



Hence, 



' f 4( 



= \/ n X 3w 2 re, a quantity manifest- 

 ly impossible, unless n be negative, that 

 is, unless two roots of the proposed cubic 

 be impossible. 



EQUATIONS, biquadratic, solution of, by 

 Des Cartes' s method. Any biquadratic may 

 be reduced to the form x*-{-g x 1 - -\-rx-\- 

 s = , by taking away the second term. 

 Suppose this to be made up of the two 

 quadratics, x 1 -f- e x +/= 0, and x 1 

 ex+g = 0, where -f e and e are made 

 the coefficients of the second terms, be- 

 cause the second term of the biquadratic 

 is wanting, that is, the sum of its roots 

 is 0. By multiplying these quadratics 

 together we have x* -f g+f e^ . x 1 -f- 

 e g e/. x T^-fg = 0, which equation is 

 made to coincide with the former by 

 equating their coefficients, or making 

 g -I-/ e 1 = q, e g e/= r, and/^ms ; 



hence, -f / = q + e 2 , also ff f - 

 and by taking the sum and difference of 

 these equations, 2 g = q -f- e 2 -f- -, and 



2/=q-j-e* ~- t therefore 4 f g = g 1 



4- 2 g e 1 + e* 



4 s, and multiply- 



ing by <? s , and arranging the terms accord- 

 ing to the dimensions of e, e 6 + 2 q e* 

 -f- q * 4 s X e 2 r 2 = 0; or making 



y =e\yi + 2 qy* + q 1 



By the solution of this cubic, a value of 

 y, and therefore of ^/ y, or e, is obtained ; 

 also/and^, which are respectively equal 



'-- q+<? 



to 



and 



, are known 



the biquadratic is thus resolved into two 

 quadratics, whose roots may be found. 



It may be observed, that whichever 

 value of y is used, the same values of x 

 are obtained. 



This solution can only be applied to 

 those cases, in which two roots of the 

 biquadratic are possible, and two impossi- 

 ble. 



Let the roots be a, b> c, a -J- b -+- c; 

 then since e, the coefficient of the second 

 term of one of the reducing- quadratics, is 

 the sum of two roots, its different values 

 are a -f 6, a -f- c, b + c, a -f b t 

 a- -h c, b + c, and the values of e 1 , or^ 

 are a + b]\ a + c]S b + ) ; all of which 

 being- possible, the cubic cannot be solved 

 by any direct method. Suppose the roots 

 of the biquadratic to be a -f b ^/ 1 ; 

 l- a + c/^T; a c 



\/ 1 ; the values of e are 2 g, b -j- c_. 

 v/ M c. ^/~^T,b^c~. +/~f, 



b + c . V' 1 and 2 a ; and the 

 three values of y arelT^V, b + c] 2 , 

 * c\ which are all possible, as in the 

 preceding case. But if the roots of the bi- 

 quadratic be a + b </ 1, a b </~l, 



a -""* fl c > the values of y are 



o , c +b^/l *,c b v/^ S two 

 of which are impossible ; therefore the 

 cubic may be solved by Cardan's rule. 



EQ.UATIOIT, annual, of the mean mo- 

 tion of the sun and moon's apogee and 

 nodes. The annual equation of the sun's 

 mean motion depends upon the excen- 

 tricity of the earth's orbit round him, and 

 is 16|| such parts, of which the mean 

 distance between the sun and the earth is 

 1000 ; whence some have called it the 

 equation of the centre, which, when great- 

 est, is 1 56' 20". 



The equation of the moon's mean mo- 

 tion is 11' 40"; of the apogee, 20' ; and 

 of its node, 9' 30". 



These four annual equations are always 

 mutually proportionable to each other ; 

 so that when any of them is at the great- 

 est, the three others will also be greatest; 

 and when one diminishes, the rest di- 

 minish in the same ratio. Wherefore the 

 annual equation of the centre of the sun 

 being given, the other three correspond- 

 ing equations will be given, so that one 

 table of the central equations will serve 

 for all. 





of a curve, is an equation 

 shewing the nature of a curve by express- 

 ing the relation between any absciss and 



