16 Proceedings of the Royal Irish Academy. 



6. In passing it is worth while to refer to possible types of sub-factor 

 other than that which it is" proposed to adopt. 



Adverting to the case of symmetry, it is known (D, § 37) that, when k- is 

 a real parameter, that branch of 



\i K -r (w 



in the relevant region of the plane of u\ which ->- 1 for ?/■-> + co is a 

 curve-factor of zero angular range, representing a straight boundary for w real 

 and < n: < r, and a curved boundary for /<• > k-. and having constant modulus 

 on the curved boundary. Hence 



(w)=Exp log[|tic + (w-/e) }lw'lf( K )dK, (4) 



J 



where f i? any function of the real variable k, is a form of ( „' which satisfies 

 the special requirements. Save for slight changes of notation and sign- 

 convention, this is tlic same as W( a "',> asdefined in 1». §40. It is mentioned 

 here because, it Mr. Levy's analysis paper E) were replaced by an equivalent 

 analysis in terms of curve-factors, it would prove tocoiTespond in the case of 

 symmetry t<> using a form of which differs from the above only as the 



product of a definite number of factors with definite indices differs from the 

 exponential of a definite integral, which is its limit when the number of 

 factors becomes great without limit and tin- index of each factor infinitesimal. 

 Similarly, Mi. Levy's treatment of the asymmetrical case would correspond 

 t.i the use of powers of sub-factors of the type 



I 2(a + e)*(/3- 1 



which is a particular case of t . and defined in F,§8. The index 



would ultimately be a function of «■ and ft, while these parameters would 

 take values respectively between - i and a, and between c and a. 



Some reason for doubting the complete generality of < „' as given by 

 formula (4 is suggested in Article 18 below. 



When Prof. Levi-Civita's analysis paper A is replaced by a parallel 

 analysis in terms of curve-factors, it is found to be essentially equivalent to 

 representing . luct of a single factor 



- :-c)i", 



in which a = /3 = », aud a number of sub-factors of the type 



; =Exp[ . . ; - o*)*j»], (6; 



n taking integral vain.--. On comparison with l». §43, it will be recognized 



