626 Mb DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



of parts, and each vertical side into some other fixed number. Let this polygon be 

 augmented in size, without alteration of form, so as to be the smallest which can have 

 all its points on the intersections of the squares, and we have the essential polygon of the 

 rationalized equation of which the given irrational equation was one of the irrational 

 forms. Or thus : — Let all the fractional exponents be reduced to a common denominator 

 k; divide each side of all the squares into k equal parts, and thus construct other squares. 

 Draw the essential polygon of the irrational equation, and then read each Ath part as 

 unity : we have thus all the essential terms of the rationalized equation. And thus the 

 character of all the solutions of an equation can be determined from any one of its 

 irrational forms, and certain a priori limitations placed upon the number and character 

 of the terms of the rational equation. 



In an inverse question, such as above described, it depends partly, but not entirely, 

 on the numerical coefficients introduced, how many of the required branches are real, 

 and how many imaginary. But the criteria for initial and terminal branches are wholly 

 distinct, no one essential term for the former being necessarily an essential term for the 

 latter ; it is even impossible that one term should be essential in both cases, except the 

 highest and lowest powers of y, when they have only one power of x in their coefficients. 

 I do not lengthen this paper by suggestions which arise relative to the classification of 

 curves, — the mode of dependence of w expressed in terms of y upon y expressed in 

 terms of w, on which every square within the polygon must tell something — the manner in 

 which the introduction of a non-essential term peculiarly affects the roots to the square of 

 which the new exponent point is a corner, &c. &c. 



To form the polygons for <p , <p l/t/ , &c. we have but to advance the axis of m one interval 

 for each differentiation, rejecting all the exponent points left behind : and similarly for 

 <p x , &c. Thus, taking the first instance in the diagram, cf> has SRX for its boundary 

 instead of SPQJf, and all the higher forms change dimension at once. The lower forms 

 of the roots are lost or modified in descending order : the higher forms in ascending order. 



The application of this method to surfaces is, so far as I have examined it, of very 

 close analogy with the preceding results : but as I can now give nothing which would 

 not easily suggest itself after what precedes, I will here only recommend it to the notice 

 of those who take particular interest in the subject. It will, I think, demand a special 

 inquiry into the theory of polyhedrons, considered as having triangular faces only or 

 else three-angled corners only; a face of more than three sides being looked upon as a 

 casual degeneracy of two or more triangles into the same plane. 



It remains to notice the pointed branch, or branche pointillce. Writers are much 

 divided on the question of admitting this part of a curve : but their controversy is one 

 of definition of terms. In geometry, it may be excluded : in algebra, it must be admitted. 

 Fully to represent y = e", we must admit every point contained in w = (2p + l) : 2q, 

 y = the negative value of ( 6 (2p+1) :? )^. In fact, e r being the real and positive value of e x , 

 we see that e* ■■ e„* {cos Zirmx + sin Zirrmv . y/ - 1 \ m being any integer, positive or 

 negative. The condition sin ferilM = includes a branch all points of which are not admis- 

 sible, but which, nevertheless, contains an infinite number of points of either character 

 between any two of the other. 



