614 



Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



theorem, and also some simplification of the demonstration, I wrote a short account of it for the 

 Quarterly Journal of Mathematics, giving the authorship to Mr Minding. Happening to 

 fall in with the volume of Liouville's journal above cited, I was surprised to find that no 

 such theorem is either enunciated or proved, but that, on the point of which it treats, reference 

 is made to Lacroix, Vol I. p. 223. This reference shews nothing to the purpose ; but taking 

 it as a hint to examine the whole volume, I found at page 102 of the introduction, a somewhat 

 prolix and uninviting account of the very theorem, which it appears was given by Lagrange 

 in the Berlin Memoirs for 1776. How completely it has dropped* out of sight will appear 

 from the uses which can be made of it, and which, it seems to me, must have been most 

 obvious to any writer on curves, or on the theory of equations, who had really obtained 

 possession of it. The theorem determines the initial and terminal branches of a curve, and 

 the character of its singular points, with much more ease and power than any method given 

 by elementary writers. It also determines the effect of infinitely small changes in the 

 coefficients of an equation upon its equal roots. But there is something still more remarkable 

 about its history : it was suggested to Lagrange by a method of Newton which is now totally 

 lost sight of, though, as will presently appear, it is difficult to see how any reader of Taylor, 

 Stirling, or Cramer, could have passed it over. 



Mr Serret has only adverted to the mode of finding descending series for the roots; 

 but the theorem applies equally to ascending series. In both the forms, it is as follows ; — 

 Given functions of x, such as A, of the form x a (a. + A'), where A' vanishes with x, or 

 vanishes with x~ x , and a is independent of x, and finite-f- both ways: this is the 

 definition of a function of the lower degree a, or of the higher degree a. Let there be 

 an equation 



Ay a + ByP + ... + Ky K + ... + Ty T + ... + Zy* m 

 where A, B, &c. are functions of x of the lower degrees a, b, &c, and a, /3, &c. ascend ; 

 or else A, B, &c. are of the higher degrees a, b, &c, and a, /3, &c. descend. It is 

 required to determine the degrees of the values of y in terms of x. The degrees a, b, &c, 

 a, /3, &c. may be integer or fractional, positive or negative: the only difference made 

 is that if a, (3, &c. differ by fractions, it will require further process to determine the 

 number of roots of each kind. Let y = x r (u+ U), and write *"(a + A'), &c. for A, &c. 

 which gives 



* I dare venture the supposition that Lagrange himself had 

 forgotten it before the Prairial of An V. It must be remem- 

 bered that every direct use of the theorie des suites was desirable 

 in the differential calculus presented as the Thiorie des 

 Fonetions: the direct treatment of singular points by series 

 alone would have been no mean triumph of the principle. The 

 reader who refers to this work (first edition, pp. 130 — 133; 

 second edition, pp. 181 — 185) will see that Lagrange reasons 

 only on the form y =fx, omitting all consideration of tj> (x, y ) m 0, 

 and then abandons the singular points under reference to 

 Cramer : he himself, twenty years before, having given the 

 means of finding a mode of development from the implicit 

 equation of a far more easy character than that which he had 

 just applied to the explicit equation. 



Lagrange's paper of 1 776 is entitled Sur Vusage des frac- 

 tions continues dans le calcul integral. The theorem is con- 

 structed upon an application to differential equations only, as 

 a means of obtaining continued fractions. Lacroix presents it 

 applied to the implicit form <j> (x,y) =0. 



+ This short phrase, which is much wanted, serves to 

 indicate that the quantity is neither infinitely small nor infi- 

 nitely great. When writers speak of a finite quantity, some 

 mean only that it is not infinite, others that it is neither 

 infinite nor nothing ; and 1 think the first meaning is the more 

 common. Thus 'a point at a finite distance from the origin* 

 usually includes the origin itself. And it ought to be so. For 

 finite and infinite ought to be contraries : and infinite, used 

 alone, always means infinitely great. 



