( 801 ) 



even and odd polynomia becomes necessary, and the general expres- 

 sion is : 



Q„ =z .i-" -\- rt^.f"— 2 _^ a^.f"— * -f . . . . a„ n even 



= A-" 4- a^.xn-- -\- OgA'"— * -{-.... an^2 n odd 



A simplification of the formulae can then be obtained by altering 

 the scale value in such a way that the limits become ± 1, w^hich 

 is always possible; for (he sake of convenience these limits have been 

 omitted in the following expressions. 



The means of different order are indicated by : 



f4,j 1= I ?^^'"c?.^■ 



In order to enable us to calculate from the infinite series (1) the 

 ^4-coeflr. in a finite form, the unique and sufficient condition is that 

 the «-coeff. be determined so that the condition : 



ƒ 



QnX^Mx^() (2) 



is satisfied for all values of m <[ n as then all integrals beyond the 

 »z + 1^'' term vanish and, at the same time, the a-coeff. are entirely 

 fixed, but for an arbitrary constant factor. 



If this operation has been performed, it is at once evident from 

 (2) that : 



ƒ< 



QmQndx =: 



for all values of m different from 7i and, further, that : 



Au = a I uQndx . (3) 



where : 



«—1=1 QnQnd.V ■= I QnX^^d.V . 



The "/q [n even) or "—72 (n odd) constants of the polynomium 

 Qn are calculated from the "/o or "~~\^2 equations : 



JQ^dx = j CQn.vdw = 



I Qnx^'dx = I (^n even) 1 Qn^v^dx = 



(n odd) 



