612 Me DE MORGAN, ON THE SINGULAR POINTS OP CURVES, 



when we contrive a conjugate or isolated point which is an evanescent oval ; as when we 

 obtain y = aty/(x 2 — l) from a = in y = y/\(x - a) (x — 2a) (a? 2 - l)\. Next, I have 

 no doubt that y has an infinite number of values ; not equivocal values, between which 

 there is choice, but simultaneous values, between which there is no choice : the sense being 

 that in which 1 — 1 + 1 — ... is and 1. And here the principle of mean values, on 

 which my reliance increases with every new case in which I try it, may place the 

 meaning of y at an isolated point in satisfactory connexion with other results, with 

 which it must stand or fall. I may perhaps be able to meet the powerful objec- 

 tion which can be urged against any application of this principle taken alone, in a 

 future communication. From examples, it seems that y = a + by/ — 1, at an isolated 

 point, means that /tan 6d9 : fdd = a + by/ — l, where 6 is the angle made by the 

 tangent of the infinitely small oval with the axis of x. The integrals are taken through 

 all the revolution of the tangent, with different signs for the contrary revolutions, if any. 

 And it seems further that when by/ —1 = 0, either (perhaps always both) of two things 

 may happen. Either the passage of tan through co may be made the same number 

 of times in each direction of revolution, so that the discontinuous constants put together 

 give k(rmry/ — 1 — miry/ — 1), or : or else axes perpendicular to some line (y = ax) 

 are infinitely small, so that y is always infinitely near to a, except at points which, 

 compared with the extent of the oval, are infinitely near those at which y = —a -1 . That 

 an isolated point may lie on a real branch, was noticed by D'Alembert : but the remark 

 is seldom repeated in modern writings: D. F. Gregory alludes to it in his Examples. 



Two methods have been adopted of treating the inquiry into the singular points 

 of curves. The first, of which the fullest development ever given is in the well-known 

 and highly-valued, but (as I shall show) little read work of Cramer '...Analyse des 

 Lignes Courbes Algebriques? Geneva, 1750, 4to, reduces the question to dependence on 

 the theory of equations, using differential coefficients, if at all, only as convenient aids 

 to development. The second endeavours to present canonical criteria, expressed in 

 terms of differential coefficients. This second method never was successful, and never 

 will be, in the full determination of points of singular curvature. No given number of 

 differentiations can in all cases discriminate, for instance, a cusp from a multiple point. 



The real dependence of the second method upon the theory of equations, when the 

 method is carried far enough, lies hid under an attempt to make the investigation one 

 of order, examining first the dependence on (p x and (p , then that on (p xx , <p xy , and <p yy , 

 and so on. When the whole progression is looked into at once, the reduction to the 

 theory of equations is easily seen. 



Let (x, y) and (x + dx, y + dy) be points on <p (x, y) = 0, and let dy = pdx. We 

 have then 



<p(x + dx, y) + d) y (m + dx, y) .pdx + d) yy {x + dx, y) + ... = 0. 



Expand in powers of dx : in each expansion preserve the lowest power of dx whose 

 coefficient has value, to determine the limit of p, or y at (x, y); and also the lowest but 

 one, to aid in examining the neighbourhood of (x, y). Suppose we thus obtain, 



