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13. Criteria of convergence for the reduction procedure in the general case 25 Expanding the right-hand sides of (23) as power series in p = x we obtain the same transformation in the form + y + ... ii ... I Xi yj . ZI ' .. 1 Y = Y + fJ ii... I x' yJ. . ', fJ', ... , y' are positive real constants, which need not be specified further. We only observe, in this respect, that: 1. equations (7') are convergent by construction; 2. ii . 1= Pii . I, (e - ek) fJ~i ... I = = Q~. }t . • , (n- nk) t;: ~ y~.
But it would not be difficult to extend finally the range of our considerations, and so give examples for any number of variables. We can add that, as a consequence of § 13, pp. 22, 26, for the required transformations the coefficients of dilatation must have absolute values neither all greater nor all less than unity; so that, in particular, for the case n = 1 any transformation having a united point at which it is not linearly representable, must have its (unique) coefficient of dilatation in modulus equal to unity.
1 Y = Y + fJ ii... I x' yJ. . ', fJ', ... , y' are positive real constants, which need not be specified further. We only observe, in this respect, that: 1. equations (7') are convergent by construction; 2. ii . 1= Pii . I, (e - ek) fJ~i ... I = = Q~. }t . • , (n- nk) t;: ~ y~. 'l,J (11~) • • • I = R~· t} . ' analogous to (11 0), with the obvious meanings for the right-hand sides. We shall show that, however the non negative integers i, j, ... ··'J . . 'J . . I, IfJ··'J . . I 1< < Tk - 1 fJ~i ...
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