442 PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



These fbrmulai are perfectly definite and intelligible, when confined to the 

 hyperbola ; and numerical values being assigned to the quantities n, x,y,c; the 

 values off {n «), 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 \/ fi, which, in this case, becomes - a/ - 1 an imaginary quantity. Still, how- 



ever, 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 x being required from the two equations 



x^ + c i/'- = a^ ; ^ ^ = b^ '■ 



we have x' + 2x yjc + cy' = a^ + 2 6Vc ; 



x' — 2xyslc-\-cy- = a^ — 2b^^c\ 



and taking the square roots, we get 



x^y.]c = ^{ce + 2 6Ve) ; x-y^c = sJ{a'-2 b'^c) ; 



and X = M V(«' + 2 6Ve) + n/(«' - 2b' ^c) } : 



Suppose now c=+l, then x=^{^{d' + 2 b') + ^{a'-2b')}; 



but if c=-l, then a:=i{V(a' + 2^V-l) + v(o'-26V-l)}. 



In the first case the value of j; is rm/, but in the second it is illusory, be- 

 cause it involves the imaginary symbol V—l. We can, however, eliminate -/— 1 : 

 thus taking the square of the expressions for a;, we have 



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

 negative. 



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

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

 xy J c( = h^ J c) from the squares of the sides of the first equation, we have 



x^-2x- y^ c + c^ y^ — a^-4:b'^c; 



and taking the square roots, 



x''-cy^ = ^(a''-4:b^c); 



* These important analytical expressions were found by De Moivbe in 1707, and inserted in the 

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

 Miscellanea Analytica, printed at London 1730. 



