d and let 



New Mechanical Quadratures. 345 



in which case w must be divisible by m. Evidently these 

 are also polygons o£ chords and there is an Eulerian equation 

 corresponding to each of them, obtainable by merely sub- 

 stituting mh for Jl. Let also 



T=2%i + 3, 3 +y { ... +^-i)=20 1 -C 2 . 



Here T may be a polygon of tangents or of tangents with 

 portions of the ordinates. It consists of n/2 portions each 

 of width 2/i. In any case the integral sought will be an area 

 intermediate between T and C m irrespective of the particular 

 value of m. It is evident from the identity T = 2C 1 — ( S 2 

 that 



4/ - • ** • 



Suppose ?i, or the number of strips into which the area is 

 divided, to be a multiple of two. Then the integral is ex- 

 pressible by each of two or more Eulerian equations. If 

 each of these is multiplied by an arbitrary multiplier and 

 if the sum of these multipliers is unity, the sum of the 

 equations will be a new expression for the integral. 

 Furthermore, for every polygon of chords involved it will 

 be possible to eliminate the coefficient of one difference of 

 derivatives, or v. Thus if n = 4t the integral may be ex- 

 pressed in terms of T, C 2 , or C 4 . Multiplying by arbitrary 

 coefficients and adding the three equations makes it possible 

 to impose three conditions : viz., that the sum of the multi- 

 pliers shall be unity and that the coefficients of v and v" 

 shall disappear. This transformation leaves the integral 

 expressed in terms of the tbree polygons and of derivatives 

 of the fifth and higher orders for which finite differences 

 may be substituted if necessary. 



So far as mere elimination is concerned there appears to 

 be no limit to this process. Thus if x n — x Q were to be 

 divided into 60 parts, eleven coefficients could be eliminated 

 and the integral would be expressed in terms of T and eleven 

 polygons of chords, only the 23rd and higher derivatives or 



