442 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



These formula' are perfectly definite and intelligible, when confined to the 

 hyperbola : and niinierical values being assigned to the quantities n, x,y,r,; the 

 values of/ (n a), and F {n ») may be expressed in real numbers : They lose, how- 

 ever, this property, when extended to the ellipse or circle, by reason of the sym- 

 bol \/ '•. which, in this case, becomes - a/ - 1 an imaginary quantity. Still, how- 



evei", they are not insignificant, for they express real quantities, although under 

 an imaginary form.* 



12. We meet with the same peculiar form of expression in the elements of 

 Algebra. Thus, the value of :r being required from the two equations 



X- + (■ y- = a' : x y = b^ : 



we have i? + 2 r y^c + cy- = a- + 2b''^c; 



r' — 1 X y^c + c y^ = a' — 2 b',Jc; 



and taking the square roots, we get 



X + y ,,/(• = V(«-' + 2 b'Jc) : x-y^c = ^{a} - 2 6Vc) ; 



and .r = i { V(«' + 2 6 Vr) + ^/(a- - 26V'-) } : 



Suppose now c=+l, then a:=i{V(«' + 2 6') + V(a°-260}; 



but if c=-l, then .r = i{V(a' + 26V-l) + V(o'-26V-l)}- 



In the first case the value of x is real, but in the second it is illusory, be- 

 cause it involves the imaginary symbol -x/— 1. We can, however, eliminate aZ-I : 

 thus taking the square of the expressions for x, we have 



«2 = i { a'i + V(«''» + )V(a'-4 6^ e) }; 



The square root of this quantity gives a real value for x, whether c be positive or 

 negative. 



The same value for x°- may, however, be found from the proposed equations 

 by proceeding in a different Avay : Thus subtracting four times the square of 

 ry^c( = />" .J c) from the squares of the sides of the first equation, we have 



x^-2x^ if r + ^^y'' = «*-46*c; 



and taking the square roots, 



• These important analytical expressions were found by De Moitbe in 1 707, and inserted in the 

 Philosophical Tnitisnctions of that year ; and again in the Trannodions for 1722. They are also in his 

 Miscellanea Analyticu, printed at London 1730. 



