20 Proceedings of the Royal h <*n Academy. 



Z real. Tt is to Vie understood that llie branch points Z = + a are excluded 

 by infinitesimal cavities from the relevant half plane of Z for --vhich the 

 function is defined. 



If dz/dZ be equal to a product of powers of functions of this type 

 multiplied by a real positive constant, the asymptotic directions of the curve 

 corresponding to Z real will he parallel to the real axis of z. 



Now let a lixed value Z„ be assigned to Z, corresponding to any point in 

 the relevant half-plane, and let a complex variable Z he substituted for k. 

 There results a function 



/•<::„> - 



-::.. ~ „>-,>■;- :-■ ir:,;--. (i3) 



and if iliis also be considered only in the relevanl half-plane of : its 

 definition can be cleared of ambiguity by postulating that the continuation 

 of the function from real values of : between the branch points o shall he 

 bj paths confined to the positive half-plane with these branch-points excluded 

 as before 



li i- important t" notice that F(Z - - ided as a function of :. has do 

 infinity "i zero in the relevant half-plane save the obvious zero at : .. 



It is proposed i" show that, ii ( (Z be any curve-factor complying with 

 the conditions set out in the first paragraph of the presenl article, then ( <Z„) 

 c.-m alv. . , product or limit-product of factors which arc 



powers of F(k, 1 i where •. take- real values from - a to a. 



I or real values of : it is convenient to put £(Q =t exp(i&) 



Tin- intern d 



| tt- 1 !.»_'/'(:. :. m/i.... / \ iu 



taken round any contour in tl ant half-plane pf f, will vanish if tiie 



contoui does not surround an infinity or branch-point of the Bubjecl of 

 integration. Cons ontoui consisting mainly of a semicircle with centre 



at the origin and large radius //. togethei with its diameter in the real axis. 

 with a semicircular detom of infinitesimal radius <- round the point c. At :„• 

 which i- assumed within this contour, log FiZ _ I has a singularity of the 

 t\|.. _ . -£); iii ordei to exclude this a circular cavity of infinitesimal 

 radius t is mad.- round :, and a cut from this across to the semicircular 



boundary. This t omplel ntour which includ.-s the two side- of the 



cut and tin- circumference ••! the cavity : and inside this contoui the subject 

 of integration has no singularities. Limit formulae are sought foi t and t 

 vanishing and R — ». It is convenient to make the cut to the negative end 

 of the diameter, as in I 



