528 



Mr DE MORGAN, ON SOME POINTS 



In biordinals, A ^ = 



$$v 



<t>a<Pbny-<Pb<Pa. 



B m -0.0 , 



"' 0.0to|,-060„|,' 



But it must be remembered that we have no right to substitute these results, containing y 

 and y , in such formulae as 3^ = \\og (\|v^ 6x - ^^J)} ,. If w e apply to <£(#, y, a, 6) = the 

 process we have applied to the more convenient form y = -^(iv, a, b), we shall find 



X* 



-M 



0/ 



)}. 



+ 0« 0»-0t0«, 

 I, 0.0M,-0i0„|/' 



where £7^, , even when {7 is a function of y', means only U x + U y . y. Thus (,ry' - t/) x|y is to 

 mean y'+ (— l).y', or 0. 



11. In the biordinal equation, the number of constants is equal to that of variables: or 

 rather, the term constant being only relative, the numbers of variables in the two compensating 

 systems are equal. Hence arise facilities for the conception* and use of a system which may 

 be called that of polar correlation, of which the well known theory of poles and polar curves 

 is a particular case. 



If there be equations involving m + n variables, of which m compensate each other, and 

 also the remaining n, and if we fix each of the m variables, and then satisfy the equations in 

 every possible way by the remaining n variables; and if, merely to give a name, we say that 

 the fixture determines a point, and the variation a curve ; we may say that the (m)-point is the 

 pole of the (w)-curve. By the converse process, we may make an (n)-point the pole of an 

 (m)-curve. And the fundamental reciprocal property is obvious, namely, that all the points of 

 any curve have polar curves which contain the pole of that curve. 



Let (a?, y) and (a, b) be two points : it is not necessary that the co-ordinates a, b, should be 

 on the same system as w, y. Let the fixed point (a, 6) be the pole of the (,ry)-curve 

 (f> (x, y, a, b) = ; and let the fixed point (a?, y) be the pole of the (a6)-curve <p (a?, y, a, b) = 0. 

 Every point of an (a??/)-curve is the pole of an (a6)-curve which passes through the pole of 

 that (#y)-curve : and vice versa. As a simple instance, suppose a? 2 + y 2 = ax + by. Let both 

 systems be rectangular, let O be the origin, and P any point. If P be (a?, y), its polar 

 (aft)-curve is a straight line through P perpendicular to OP : if P be (a, b), its polar 



* When I first wrote this paper, in July, 1853, I did not 

 know that Professor Boole had distinctly conceived and used 

 this system : see the Cambridge and Dublin Journal, vol. vii., 

 p. 156. Our lines of application, however, differ widely. Mr 



Boole, as he states, was equally unaware that the subject of 

 polarity, in as general a point of view, had been presented by 

 Mr Druckenmuller, in Crelle's journal. 



