324 



REPORT 1865. 



+ CO 



1|O O)= 2„, 2 i< 2 "'+ 1 > 2 cos(2m+l) 



— CO 



a 



CO 



=2 2 i cos !* n,„(l-2 2 '") cos nJl+2^" t^ + 2 4 »'\ 



« 1 1 \ a J 



(11) 



-I + co *. 



i 0j 1 (x) = 2 (-l)^ 2 "^' sin (2m+l) — 



1 ' -co a 



=2<z* sin ^ n,„ (l-<f") n m f l-2f/» cos ~f +rf ; . (12) 

 a i i \ a J 



by which the Theta functions are expressed as convergent products of an in- 

 finite number of factors. 



Other important consequences are deducible from the equation 



2 <W vx Oi) M 2 , v* (*J e M3. «3 (*») K, *4 (*J 



= <r _ Ml , „/_„, (s-a-J x 0„_ M2> <,■_„ (s-a?J x 6^^ „,_,,., (s-.r 3 ) 



+ <r _ ( u 1| *•_>,+! (S— a\) X fl,,-^, tr'-vi+l («—<*„) X 0<r- M 3> •*-*+! («-<> 



+ ( — 1)'' 0<r- Ml + l, «r'-*, ( S -^i) X 0O-M.+ 1, a»-i* (s-*a) 



X flo-na+l, <,'-*, (*— * 3 ) X Oa-^+l, <r'- Vi (S—Xj 

 + (-l)l+^0 <r _ Ml + 1) ^_, 1 + l (S-^) X 0C-M.2+1, <r'-,,+ l (S— * a ) 



X 0<r-h+I.o'-ib+l (*— #s) x Oa-^+L^-K.+ i ( 8 —®J> 

 which contains four independent arguments, x 1 x 2 x 3 x i , and in which 2s = 

 a7 1 +a? a +ar a +i» 1 , 2»=/u 1 +/i i ,+/i,+/u 4 , 2<r'=r 1 + v 2 + v 3 + ^; the numbers 

 fi l fx i fx 3 n i and v, Vjj^j'.j being subject to the restriction that their sums are 

 respectively even, so that a and a are integral*. Let V k, V k' be two quan- 

 tities defined by the equations 



v;_^. rtJM&s (14) 



attributing in (13) to the elements 



the systems of values 



(i) 



Hv /V P 3 > lh 



0, 0, 0, 

 0, 0, 0, 

 0, 0, 0, 



* This very symmetrical formula is, it would seem, nearly the same as that employed by 

 Jacobi in his Lectures on Elliptic Functions at the University of Konigsberg (see his letter 

 to M. Hermite in Crelle's Journal, vol. xxxii. p. 177). It may be proved by actually 



tir 



multiplying the four Theta series, and transforming the indices of —1, e™, and e* in the 

 general term of the product by means of the elementary formula; 



a*+b(3+cy+dS=(s-a)(2- a )+(s-b)(2-P)+{s-c){V-y)+(s-c!)(2-S), 

 where 2s=,a+b+c+d, 22 = «+ / 8 + y+£. 



