310 Cambridge Philosophical Society. 



equal roots, what effect will be produced upon these roots by given 

 infinitesimal alterations in the coefficients, how many will remain 

 real, and how many will become imaginary ? 



Newton has given the foundation and the chief step of a geome- 

 trical method (Neivton s parallelogram) which has passed into oblivion, 

 though it occurs in the celebrated second letter to Oldenburg, has 

 been fully described by Stirling, used by Taylor and De Gua, and 

 forms the main method of Cramer's work on curves. Mr. De Mor- 

 gan proposes to call it the method of co-ordinated exponents. 



He proceeds to describe and enlarge this method ; observing that, 

 of the polygon which represents an equation, Newton and his fol- 

 lowers are in full possession of the connexion of the sides with the 

 solutions, and fail onlj"^ in not grasping the connexion of the whole 

 polygon with the whole equation. Both Newton's method and 

 Lagrange's, the second of which is an arithmetical version of the 

 first, may be applied to irrational equations, but it will be convenient 

 to confine the description to the form ^ax''"y"=-0, where m and n 

 are integers. 



In ax'^i/'\ let n be an abscissa, and m an ordinate, and let (m, n) 

 be called the exponent point of the term ax^i/". Take some paper 

 ruled in squares (or ruled both ways in anj^ manner, for any equal 

 rectangles will do) to facilitate the process when n and m are always 

 integers, and lay down all the exponent points in 2!aa''"j/"=0. 

 Through some of these points draw a convex polygon including all 

 the rest, which can only be done in one way. Should the points be 

 so many and so scattered that some method must be applied, the 

 geometrical method is a translation of the main arithmetical method 

 of Lagrange's theorem. The points which end on, or otherwise 

 fall in, the sides of the polygon show the essential terms of the equa- 

 tion : no others are wanted to determine q and u in y=a;''(w-|-U). 

 The upper contour of the polygon shows how all the solutions com- 

 mence in descending powers of x ; the under contour does the same 

 for ascending powers. Take any side of either contour, its projec- 

 tion on the axis of n shows the number of roots it represents, the 

 tangent of the angle it makes with the negative side of the axis of n 

 shows the value of r. 



It will not be needful to abstract the developments given in the 

 paper : we shall only notice the inverse method. The following 

 example is taken, and the construction of the equation is even easier 

 (under Cramer's form) than the direct treatment of it. The example 

 chosen by the author is the following : — Required (b{x, y) = 0, of the 

 twelfth dimension in terms of y, such that the twelve roots of y, 

 with reference to lower degrees, shall be as follows : two roots of 

 the degree 1, four of \, two of 0, one of — 1, two of — %, one of — 2. 

 But with reference to higher degrees, there are to be one root of the 

 degree 3, two of \, three of 0, three of — i, two of —1, one of —3. 

 On examination these conditions are found compatible, and the most 

 general equation which satisfies the conditions is found. 



The paper is terminated by a discussion on the pointed branch, 

 for the admission of which, as a branch altogether composed of sin- 

 gular points, the author contends. 



