[108] PROFESSOR DE MORGAN ON SOME 



It is possible that either A or B may contain a factor which is a function of a or b, or 

 both, and not of m nor of y. By making A or B, whichever it may be, vanish by this factor, 

 and the other by a relation between x, y, a, and b, we obtain from the two equations a real 

 family of curves, satisfying = 0. And B always has some positive power of b for a factor, 

 which gives the most common case of this kind. Moreover, it is possible for an isolated ima- 

 ginary value of c to give a curve : as in x 2 + (e 2 + c) y — c (c 2 + c + l) which gives continuous 

 curves for values of c determined by c 2 + c + 1 = 0, and for no other imaginary values. 



If ever it happen that a continuous series of imaginary values of c make (p (x, y, c) = 

 the equation of a continuous series of curves, it is a sign that this equation can be thrown 

 into the form \js (x, y, •arc) = 0, where the imaginary values of c give real values to •arc. 

 For it is certain that the equation of a real family of curves can always be thrown into the 

 form y{/ (x, y, k) = 0, in which k is real. 



Since <p (a?, y, c) = gives values of c, real or imaginary, for every value of x and y, it 

 follows that every point in the plane of xy is either a point in a continuous curve of solution, 

 or an isolated point, or both : and it may be either or both of these in several different ways. 

 As the point changes place, a critical pair of values of x and y may arise, at which the cir- 

 cumstances alter : an infinitely small change may bring us, for instance, from a point in six 

 ways isolated and ten ways connected, to one in four ways isolated and twelve ways connected. 

 Let c = a + \/b, and, cp meaning <p (x, y, a), and <p a <p aa , &c , its differential coefficients with 

 respect to a, let Taylor's theorem apply. We have then 



(p(x,y, a + \/b) = $ + (pn, - + ...+ ^b.((p a + cp aaa — + ...). 



If b be very small and negative, the conditions under which the equation can be satisfied are 

 <p = °j <p a = °> nearly. Accordingly, if a be eliminated between (f> = 0, <p a = 0, we have the 

 equation of the curve at which the changes of character take place, whenever they happen 

 by b passing through zero, under values to which Taylor's theorem will apply as far as d> aaa 

 inclusive. 



For example, let the given equation be y — fc —fc(x — c) =0, in which the second 

 equation must be /" c . {x - c) = 0. All the straight lines in this family are tangents to 

 y =fx, and we know that passage of a point over a curve generally makes the number of 

 tangents which can be drawn through the point greater or less by an even number. But 

 f"c = may indicate a point of contrary flexure : and a little consideration will shew that 

 passage over any part of the tangent at a point of contrary flexure alters the number of 

 tangents which can be drawn through a point. 



When the modern analysis first impressed mathematicians with a sense of its power, it 

 would seem as if they accepted it as fit to give results of the same extent as those of 

 geometry. Perhaps they forgot that geometry makes its apparent universality of conclusion 

 valid by preliminary restrictions. Euclid excludes one singular case of angular magnitude, 

 the angle of a straight line and its continuation, altogether : and the extreme case of inter- 

 section of two lines is excluded by referring it to a separate name, parallelism. Hence it is 

 that two sides of a triangle are always together greater than the third ; and that two inter- 



