106 REPORT 1870. 



^-^(^')=^V^^^'' ^-(■^■)=''^(?') • . • . (3) 



'-(•^■)= a;^ • «3 ^'. eo,oi-)=e('^) .... (4) 



This notaliou allows us to make use of the following definition, which is 

 of fundamental imiwrtanco throughout Professor Betti's memoir : — 



0U^') = ^^--1^y^^ \ ^ \-±ll, ... (5) 



when 



K,(^.+-^'.') 



~ — (2.r:-\-fMi-\-io') 



also 



e„.(.)=e"^" ~V,»(.+9^+f) , . (7) 



Section 6. — Having thus explained the notation, we come to the following 

 theorem given by Professor Betti (A. D. M. 3. 123) : — 



Po.o(z) + (-irPoa(^) + Pi.o(--) + (-irPM(^), .... (1) 

 where 



P», e (-) = 0/x + r,, v + e (~) ©m' + I, c'+eC-) 6,^_^'+„, ^ (?(') 0„, ^_^' + , (m/). 



The roots of the entire functions Q^^^{z+tv) e^>.(;:— ty) are respectively 

 of the form 



_^, + (2.+^_l)^+(2s+,_l)|, 



and the theorem is shown to depend on the proposition that these are also * 

 roots of the expression 



F(.)=p„,,(.)+(-irp„.,(.)+p^.x~-)+(-irp.,« . . (2) 



which it appears wiU be true if 



Q^-f.'+hv-p'+ii'M) ©I, i(w) G^_^> («') eo,„-^' (iv), 



+ er,-^',p-r,' + l(tv) 00,1 iy 0^-i^' + i,o(w) 01,^-;,. (iv), 1 



— e^_^'+,, y _„<!(') 0,,on' 0^-^i',i («') eo,i,-„'+i('U') m 



— 0,x-^.', r-..' Of) 00, «(*'') Q^-^' + i,i(u') 0i,^_^' + i 10 = 0. 



The reader will find no difficulty in proving this by means of the formula 

 of last section, and the expressions for the periods given by Sehellbach, p. 34. 



i 



