AND ON NEWTON'S METHOD OF CO-ORDINATED EXPONENTS. 625 



we pick out the essential terms by help of the polygons, and proceeding as above directed, 

 we have the following ; 



3 1+5 7 1+7 - - 1+2 



2 + 5 8-9 1 - 1+6 



5 + 8 3 + 10 9 + 12 3+12 - - 3 + 7 



11 9 + 13 15 9 + 15 - - 9+10 



12 10 + 14 16 10+16 - - 10 + 11 



3 2+6 8 2+8 - - 2+3 



4 + 9 2 + 11 8+13 2 + 13 - - 2 + 8 



5 + 8 3 + 11 9 + 13 3 + 15 3 + 15 2+l6 3 + 16 2 + 11 8 + 13 2 + 13 2 + 3 - 2 + 8 



Reckoning from the lowest left-hand corner, the sides of the first polygon may be 

 signified by A (horizontal), B, C + D (in one straight line), E, F, G (vertical) : and those 

 of the second by I (horizontal), II, III (vertical), IV, V, VI, VII (vertical). The 

 polygon belonging to the product has then 14 sides as follows; — A+I (in one straight 

 line), II, B, III, IV, (C-r D), V, E, VI, F, (VII + G). This order is plainly determined 

 by the slopes of the sides. Through one point draw lines equal and parallel to all the 

 sides of both polygons in the directions in which we pass over the several sides in making 

 the circuits : the order of intermixture thus established shews the order of the sides 

 in the compound polygon. This would be, on ruled paper, the easiest way of making 

 the multiplication. Also, by taking note of the projections on both axes, in the pencil 

 of the polygon, we may facilitate the inquiry whether a given expression is reducible : 

 if it be so, the pencil can be divided into two systems of rays, in each of which the sum 

 of both projections is zero. It depends on the coefficients whether the reduction can be 

 completed : but in every case in which the polygon is reducible into other polygons, 

 there must be a certain reducibility of form in the function, and a certain divarication 

 of the roots, which does not belong to expressions in general. 



Returning to the original question, if in the product we destroy the highest term by 

 introduction of a coefficient 0, we lose one root B on the lower contour, and one root IV 

 on the higher : these are certainly not the same. 



In treating of the roots of an equation, of a degree higher than the fourth, whatever 

 we may think probable, we have no right absolutely to assume that the roots formulize. 

 My meaning is as follows; — Let there be an equation of the second degree, one coefficient 

 of which is a, and the roots m and n. Alter a into A, which changes the roots, say, 

 into M and N. We can lay our hand on one of the two, M or N, and say, this is the 

 root into which change of a into A has converted m. But if we pretend to say as much, 

 or to suppose as much must be predicable, in the case of an equation of the fifth degree, 

 we assume the qualities of formular expression to belong to results of which we know 

 nothing except, at most, that algebraic formulae do not exist. 



When an equation has irrational terms, the whole process is in no respect altered. 

 If no exponents enter except definite fractions, the essential polygon of the curve will 

 have some or all of its points, not on the intersections of the squares, but on the inter- 

 sections of rectangles made by subdividing each horizontal side into some one fixed number 



80—2 



