901 



if z be taken as a complex number it represents for any value of 

 X in the domain («), ra>d for arbitrary values of z, a convergent 

 power-series in Zy and consequently also a function of z regular at 

 ^ = 0; the name operative-inixQXion is justified by this. The reason 

 of it is at once clear if we notice that the peculiar character of the 

 complete transmutation consists in the coefficients am {x) being for 

 any value of x in the domain of completeness («) less in absolute 

 value than the m^^ power of a positive number a{u) independent 

 of m. If we indicate the operative-series belonging to Pj and P, 

 respectively by f^ {x,z) and ƒ, {x,z) or simply by f\ and ƒ,, ƒ, and 

 ƒ, are regular functions of z in a neighbourhood of z ^Q, if x 

 is a point of the domain («). Since the resulting series P and P, 

 resp. arising from the schemes (56) and (56), are as already found, 

 both complete in the domain (<t), the corresponding. operative-series 

 f{x,z) and F{x,z) will further for any value of x in the domain 

 {'(), converge as well in a neighbourhood of ^ = 0, and represent 

 there a regular function of z. 



These operative-series might also have been obtained more directly 

 from the schemes in question, by tirst replacing Dhi {D^'u) everywhere 

 by z^, and afterwards collecting all the terms with ^^ as a factor. This 

 term is indeed the one to be substituted for the term with the factor 

 D^u {D^u) in the series P (P) found above; for, the latter had 

 arisen by collecting all the terms in the original scheme with Dhi 

 (D^u) as a factor, and all these terms and no others pass into terms 

 with z^. Since now, in consequence of the distributive property of 

 multiplication, the resulting coefficients both of D'^u {Dhi) and z^, 

 are equal to the sum of all separate coefficients and by the substi- 

 tution in question the coefficients remain unchanged, the two resulting 

 coefficients are equal, and consequently the sum of the terms to be 

 substituted for the terms with the same power of Dhi {Dhi), is 

 equal to the term to be substituted for their sum, as was to be proved. 



The substitution in question makes the first scheme pass into 



'-'' ^ d.x dz ^ 2! d.^' 02' ^ 



[ 



a. in w , a(;.,.)ö/ 1 dV-,z)d% 



d(;i,^^)d/, 1 ö'(A,2^)öy, 



Oa; dz 2/ da; o^ 



(58) 



