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



621 



choose from among them the corners of a convex polygon which contains all the rest within it : 

 any side of that polygon being produced, all the points, except the two or more which lie in 

 that side, are either all above the line of the side, or all below it. To determine this 

 containing polygon, take the lowest* value of n, and the lowest corresponding value of m, 

 and, drawing lines from (», m) thus taken to all the other points, proceed upon the lowest of 

 those lines, that is, upon the line of greatest slope downwards, or of least slope upwards. 

 Thus is determined the first side of the lower part of the polygon. Proceed upon the side 

 determined to its end, that is, to the last of the points, if more than two, which lie upon it. 

 From this last point repeat the process, neglecting all the points which have been passed, that 

 is, through whose values of n we have passed. We thus get a second line, and so on : in time 

 we must arrive at a point having the greatest value of n, and the lowest value of m which 

 goes with it. We have thus completed the lower contour of the polygon. Proceeding from 

 the last point obtained, which has the highest value of n, draw a vertical from the lowest 

 value of m to the highest, if more than one value of m belong to this highest value of n : this 

 vertical is one of the sides of the polygon. Proceed from this highest point backwards, and 

 draw lines to all the others, retaining only the line of greatest slope upwards, or of least slope 

 downwards : reverse, in fact, the preceding process. We thus obtain the higher contour of 

 the polygon, and finally reach the point we began from, reaching it at last by a vertical side 

 downwards, if necessary. 



The process of Lagrange's theorem will be easily seen in the above. If (n, m) be the 

 first point, and (ri, m) another, the downward slope of the joining line is more or less 

 according as (rri — rri) : (ri — ri) is lower or higher, that is, according as (m - rri) : (ri - n) 

 is higher or lower: and the rule for finding the greatest slope downwards, in proceeding 

 forwards, is also the rule for finding the greatest slope upwards, in proceeding backwards. 



The external points of the containing polygon, whether they terminate sides or lie within 

 sides, are the exponent points of the essential terms of the equation ; meaning, the terms on 

 which the characters of the solutions of y in terms of x depend, or on which the initial and 

 terminal characters of the branches depend. The lower contour of the polygon settles the 

 expansions in ascending powers : the upper contour in descending powers. To shew this, let 

 (n, rri), (ri, rri), (n", rri') lie on a side of the lower contour : we can then take p so that 

 nt + np = rri + rip = rri' + ri' p. Every other point is above the line passing through these 

 three points: so that, (ri" , rri") being any fourth point, m"'+ri"p is higher than m + np, &c. 

 Consequently, y = x p (« + U) substituted in the equation, gives three terms of the degree 

 m + rip, and all the others of higher degree. And similarly for any side of the upper contour. 



Cramer (and the same may be said of his predecessors) does not grapple with the 

 whole polygon, but takes one or another side, as wanted : the sides - )- being obtained tenta- 

 tively by help of a ruler. This method pervades the whole of his book, and constitutes its 



* By lower and higher I mean algebraically less and greater: 

 thus — 10 is lower than —1. It would perhaps be convenient 

 if lower and higher were reserved for this purpose, and greater 

 and Jess were allowed to retain their arithmetical meaning, 

 which, for the most part, they do. Thus when we say, Let ae 

 be infinitely less than y, we never mean, Let x -co . The 



words lower and higher might be used when we proceed from 

 — es to + oo , while previous and subsequent might be used, 

 as convenient, either when we proceed from — oo to + oo, or 

 from + oo to — oo. 



+ This ruler was wanted because the squares, or cases, of 

 Newton's parallelogram were supposed to be filled up with all 



