624 



Mr DE MORGAN, ON THE SINGULAR POINTS OF CURVES, 



The values of the coefficients being written* at the exponent points, we see imme- 

 diately how to write down the equations for the determination of u. This question 

 remains ; — The roots determined by the upper contour being identical, each to each, 

 with those determined by the lower contour, how are we to equate them ? Denote 

 the roots by the sides of the polygon; thus we may say there is one root IH and three 

 roots ML. If we can take off a slice from either end by a vertical diagonal joining two 

 essential points, it seems clear that the roots thrown away above and below are identical. 

 Bound the polygon by GI, instead of GH, HZ: eleven roots are untouched in dimension, 

 and in value of u : the root GH is identical with HI. Carry on this process, and we 

 see that the roots in FG are those in KI, that in EF is that in sK, those in LS are 

 those in DE, those in CD are those in uM and ML, pC is Nu, and BP is BN. This 

 seems clear, and many simple instances confirm it so far as they go : but it is not 

 proved, and certainly is not true in reducible functions. If the polygon admit of 

 formation into two or more polygons by use of sides equal and parallel to the several 

 sides (that is, if the system of balancing forces be composed of several such systems) we 

 have the case which may give <p(x, y) reducible into real factors, and without which 

 it cannot be so. Take forms of (p(x, y) from different polygons and multiply them 

 together, the sides of the simple polygons will be intermixed in the compound polygon, and, 

 since each simple polygon contains two sets of roots, identical each to each, the theorem 

 cannot be universally true for reducible equations. And since an infinitely small 

 alteration in a constant will change a reducible expression into an irreducible one, it 

 seems proved that the theorem cannot be universally true even for irreducible expressions. 

 The mode of pairing the roots of the upper and lower contour seems to depend upon all 

 the roots, at least with exception of certain cases. 



When we multiply two expressions together, essential terms in the product are 

 products of essential terms in the factors; but the converse is not true. Having paper 

 ruled both ways, the simplest plan of detecting the essential part of the product will be 

 to write down dimensions only of the coefficients in x, in columns devoted to the powers 

 of y, substituting some sign of blankness for missing powers, since is appropriated 

 to the function of no degree. The form of algebraic multiplication may then be used, 

 the component processes being only additions [of exponents]. Essential terms only are 

 retained in the factors, and in the partial multiplications only highest and lowest powers 

 need be retained : while the final process is selection of lowest and highest in the columns. 

 Thus, having to multiply together 



a?y + (a? + x b ) y b + (a? 4 + x') y* + (a? + x~) y 3 + x*y 2 + (a?* + x a ) y + « + x~ 



(a? 2 + x 5 ) y 6 + (a? 1 + X s ) if + (x 3 + a? 4 ) y* + (a? 5 + a/ 9 ) y* + (a? + x*) y 2 + (a. 2 + a; 3 ) y + x+ x* 



* This is done in one of the diagrams. If the coefficients be 

 thus written, the whole equation is written down, and it would 

 seem that, for many purposes, there is no way so good of 

 writing down an equation. Nothing is wanted but a supply of 

 paper ruled both ways. And here it must be remarked that for 



all matters connected with intersections, angular gradations 

 (not magnitudes) and general figure of curves, the paper need 

 not be ruled in squares : equal rectangles, or even equal oblique 

 parallelograms, will do as well. 



