ON THE THEORY OF NUMBERS. 323 



WW =(-i> e „.,(«0 (i) 



<V, .-+>(*) = m,»» (2) 



»„,<-•) =(-l)^^,(^) (3) 



»*„(*+•* =(-~iy ^,v(«) w 



^ fi ,(^+ aw )=(-l)- fl ftF (»)r-K»T+-) (5) 



0^+^, + ,<^)=^ > ,(^+KA + »'')«)X6^'1+^ 3 t-t]. • (6) 

 Thus there are only four different Theta functions, OO (x), O1 (jb), 1>o (x), 

 0i,i i x ) (equations 1 and 2) ; of these, the first three are even functions, the 

 last ah uneven function (equation 3) ; they are all periodic, having a or 2a 

 for their period, according as /x is even or uneven (equation 4) ; the quotient 



IXV , [ is doubly periodic, having aw or 2auj for its second period, according 



as v— v is even or uneven (equation 5) ; finally, any one of the four can be 

 expressed as the product of any other by an exponential factor (equation 6). 

 The identical equations 



l + q(v + v- 1 ) + qXv 2 + v- 2 ) + q\v 3 + v- 3 ) + q'%v i +v-')+ ■> 



Kl-tfXl-^Xl-S 6 )*' • • .x(l+2</>(H- 2 V)(l+ 2 V) .... I . (7) 



X(l + qv- 1 )(l + ^v- 1 )(l+^ir 1 ) J 



q*(v + v- 1 ) + qS(v 3 +v- s ) + qT(v 5 + v~ 5 )+ .... "i 



=(l-<f )(l-2*)(l- ? 6 ) .... X qKv + v- 1 ) I 



x(i+2V)(i+ ? ^ 2 xi+?v) /••••(»; 



X (1 + q 2 v-*Xl + 2 ^- 2 )(l + q°v-*) . . . ., J 



in which v is any quantity whatever, and q any quantity of which the ana- 

 lytical modulus is inferior to unity, express an important property of the 

 Theta functions. Elementary demonstrations of the first have been given by 

 Jacobi and Cauchy* ; the second is immediately deducible from it, by writing 

 qv 2 for v, and multiplying by q*v. We infer from these identities the four 

 formulae 



0o.o(*)= 2 m 2""cos^If 

 -oo a 



= n m (1 -q*») n„,( 1 + 2<f' - 1 cos S^+2 4 "- 2 ) ; ... (9) 



1 1 v a / 



+ °° 9 . 



d 0ll (x)= 2 (-!)■» g»» COS ""*** 



-oo « 



= n w (i- 2 2 ™)n m M-2 2 2 »- 1 cos^4Y n '- 2 ); • • . (10) 



* Jacobi, Fundamenta Nova, p. 176-183 ; Crelle's Journal, Vol. xxxvi. p. 75 ; Cauchy, 

 Comptes Rendus. vol. xvii. p. 523, 567. See also the note (by M. Herniite), ' Sur laTheorie 

 des Fonctions Elliptiques' in the 6th edition (Paris, 18G2) of Lacroix, Traite Elenientaire 

 du Calcul Differentiel, vol. ii. p. 397. 



z 2 



