3i0 REPORT— 1869. 



Section 2. — It will be proper, on every account, to commence this division 

 ■with an account of Jacobi's later researches relative to fimction 9. Three of 

 these are contained in the 36th volume of Crelle's Journal, and I shall con- 

 sider them ill succession. The first of these is entitled " Uber die unmit- 

 telbare Verification einer Fundamental-Formel der Theorie der elliptischeu 

 Functionen." 



It is proved in the ' Fundamenta Nova ' that if 



n=(l-2=)(l-2^)...(l-2--)(l-fA)...(l-2.-')(l-2z-')... 

 S and n are equal, and we must therefore have 



rfS _ i,d log n rfS _ o fHog II 

 dq dq ' dz dz 



It appears, without much difficulty, that if (m) represent every whole 

 number from 1 to co , xpim) the sum of the factors of (m), jj all odd numbers 

 from 1 to infinity, i all whole numbers from -co to +co , these equations lead 

 to the following : 



where SS refer to m and jo as above explained. The object of the paper is 

 to prove these formulae, and I make the two following observations to faci- 

 litate its study. 



The first observation, in which S refers to p, is intended to illustrate 

 page 77. 



/ (-1)">2'"^'"^''~"'' + S" (-iy'2yq'^^'''+P+^'^ 

 =2""' (-l)'"(2i + 7r)2'"<'"+"^ + 2" (-l)"'(4i + 2p)j92'"<'"+'P+''). 



— {2i 1) 



The second refers to page 78 ; there we have the following equation — 



SS( — 1) P2 =I,4^(m)q . 



It would be better to write this in the following way — 



SS(-l) pq =^^(>-)q , 



since m(m +jj) and 2r are intended to be equivalent. 



Section 3. — Jacobi's second paper is entitled " Ueber die partielle Differential 

 gleichiing welcher die Zahler und Nenner der elliptischen Function Genuge- 

 leisten." Jacobi at the commencement of his paper gives the following for- 

 mula without demonstration : — 



0a('^O 



=\/v 



2K, y„ EWi«-i_- 



Ae"' 



