530 



Mr DE MORGAN, ON SOME POINTS 



primordinals of (x, y, a, (5) = 0. And since y = fx makes A, B, determinate functions of x, 

 we see that the problem is solved for every way in which v {A, B) = F/j. (A, E) is capable of 

 being identically satisfied. Assume the form /x, then Fft (A, B) becomes a determinate func- 

 tion of x, say kw; and v (A, B) = kx can then be satisfied in an infinite number of ways. 

 Choosing one of these, we have m = /j. (a, 6), n = v (a, b) from which to determine a and /3, 

 and (<r, y, a, (3) = is a modular equation to which y = fx, n = Fm are polar reciprocals. 

 Accordingly, we can in every case dictate one of the two species of reciprocal curves, subject to 

 the usual difficulties of extreme cases. 



13. The primitive (a?, y, a, b) = gives two correlative biordinal equations, say 

 y" = j£ (x, y, y) and b" = a (a, b, b'). And we have 



whence ($ 10) A^fy = X b ,<p y , and the condition of primordinal singular solution is the same 

 both for y"=x an< ^ b"=a. Let this condition be <j (x, y, a, b) = 0. 



cr(x, y, a,b) = 



(p (x, y,a,b) = 0. 



<p* + f»y 



<Pa + W = 



a(x, y,a,b) = 



y = w («, y) 

 I 



y = n {x, c) 



a=A(x, y, y) b = B (x, y, y) x = X{a, b, b') y = Y{a, b, b') 

 I I I I 



sT=x(»,y,sO 



b" = a (a, b, b') 



b'=\(fl,b) 



I 

 b = A (a, Z) 



If we eliminate a and b between (p = 0, a = 0, (p x \ y = 0, we deduce y =•&, y = II, the 

 singular primordinal and singular primitive of y" '= ^. If we eliminate a; and y between 

 = 0, a = 0, <p a , b = 0, we have ft' = \, fc = A, the same of 6" = a. Each singular primitive 

 of y" = j£, for each value of C, is the (*y)-curve which touches all in cp \x, y, a, A (a, Z)\ = 

 for one value of Z and every value of a : and each singular primitive of b" = a, for each value 

 of Z, is the (a6)-curve which touches all contained in {x, II (a?, C), a, 6} for one value of C 

 and every value of x. The contacts are of the second order, and the ordinary primitives of 

 y =w and b'=\ are polar correlatives, each to each. This is not necessarily the case with 

 the singular primitives, which are determined from <p a = 0, &c. independently of y' and b'. 

 But if it should happen that the singular primitives are polar correlatives, it follows that they 

 solve the biordinals, and have contacts of the third order with original primitives. The proof 

 of this is in my last paper, at the end of the first section: but the result is not correctly 

 stated. The contact there mentioned is of the third order, if as high as the second : the 

 closest primitives touch the singular primitive of the singular primordinal with a contact of 

 the first or third order, not of the second at most. 



The above statements contain the assertion that = 0, a = 0, y = II, b = A, containing 

 x, y, a, b, C, Z, though only four equations, admit of the elimination of the four quantities 

 x, y, a, b, and lead to a relation between C and Z only. To show this, throw y = II, b = A, 

 into the forms C = C(x,y), Z = Z (a, b) : then if for a and 6 we substitute in terms of 



