G. F. Becker — New Mechanical Quadratures. 119 



Euler's formula is capable of some transformations which do 

 not seem to have been noticed and a variety of special forms 

 can be deduced from it. For this purpose it is convenient to 

 make certain changes in notation. The first term of the 

 second member is a polygon bounded by the axis, the extreme 

 ordinates and chords connecting the extremities of all the 

 ordinates. Let this polygon of chords be denoted by C, and 

 let 



0. 



= 2a(| 



+ y*+y*+ ••• + 



2/A 



2/ 



= mh(^ +y n 



+ 2A m +--- + 



f) 



in which case n must be divisible by in. Evidently these are 

 also polygons of ehords and there is an Eulerian equation corre- 

 sponding to each of them, obtainable by merely substituting 

 mh for A. Let also 



T = 2h (y 1+ y 2 + y 5 + . . . y^) = 20, - C a . 



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 — C 2 that 



OCii 



/■ 



ydx = T + (2* 



. 2) ?£V_ (2 ._ 2) W + 



2! 



4! 



+ R. 



Suppose n, or the number of strips into which the area is 

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

 sible 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 = 4 

 the integral may be expressed in terms of T, C 2 , or 4 . Multi- 

 plying by arbitrary coefficients and adding the three equa- 

 tions makes it possible to impose three conditions, viz. : that 

 the sum of the multipliers shall be unity and that the coeffi- 

 cients of v' and v'" shall disappear. This transformation 

 leaves the integral expressed in terms of the three 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 a — x were to be divided 



