[harkness] study of ELLIPTIC FUNCTIONS 93 



number is an integer (positive or negative) we shall use p. The 

 context will show when the numbers are supposed to include o among 

 their possible values. 



A system of periods has the property that if fij, U2, • • -, ft, 

 are members of the system, so too is pj flj+p, fl^H hPr Qr- 



Select two periods fij, Q,^ from the system. Either they are or 

 they are not connected by a relation m^ O^ + m, îîj = o- I^ they 

 are not so connected any third period fij satisfies in conjunction with 

 Qi, ^2; ^ relation 



m I fij + nij ^2 + 11^3 ^i = o 



Thus all periods of the system are of the form m\ fi^ + m'2 Î22, and 

 each period is expressible in this form in only one way. 



If the periods fij, fij are connected by a linear relation, there 

 may be some pair in the system whose members are not so connected. 

 Take these as fli, Î2^ and apply the previous reasoning to them. 

 Any system for which such a pair exists will be called normal; if there 

 is no such pair the system will be called special. The former system 

 may be denoted by [ flj, ^2 ]! the latter by [ fi ]. 



All periods of a normal system are of the form m^ 0^ + m2 Q2 

 and the combination can only vanish when mi=m,=o. All periods 

 of a special system are of the form m S2. 



Any system of periods contains infinitely many elements. It may 

 happen that the number of periods whose absolute values are less than 

 G, where G is an arbitrary positive number, is finite. Let us make 

 this hj^pothesis, and let us find what it implies in the normal case. It 

 v/ill follow that every period ^3 is of the form 



and that if Çî^, flj is replaced by a suitably chosen pair ^^, ^2, the 

 system is given by p^ Jij + P2 U.- The pair 2^, ^^ is called a 

 primitive pair, and the number of such pairs is infinite. 



The method of proof used by Weieistrass for the more general case of 

 systems of simultaneous periods for functions of n variables is available. 



Select a period flj so that either (a) o < m ^ 1, nig = o; or (b) 

 o^mj < 1,0 < m2^1. Such a selection is possible, for Î2,, ^2 them- 

 selves satisfy the 'requirement, as is obvious from 1^^= 1 • Oi + o • 0, 

 and ^2 = o • ^1 + 1 • ^j- The limitations on nij, va^ show [ ÎÎ3 ] 

 is < a suitably chosen G. All that is necessary is to show that 

 I mj fii + nij fl. I has a finite upper limit, when ra^, m., are treated as 

 variables ranging from o to 1. [The points that result in this way fill the 

 parallelogram of periods whose corners are 0, fli, fl^» ^i + ^2-] It 

 follows that the hypothesis in the theorem is satisfied. There is then a 

 period for which m^ in (a) has a least value, say mj,j; also a period for 



