94 THE ROYAL SOCIETY OF CANADA 



which nij in (b) has a least value, say m,,,- We have then, periods 

 fii, «2, where 



fii = nii.i U^, 



Î2 == m._,,i fi, + m,,, Q.,. 



The expression m^ fij + m2 S2.>, in which o ^ m, < nij,!, ; o <. 

 m 2 < m 2, 2 cannot represent a period unless nij = m, = o. 



We have now to show that the m's in fl.j=mi 12 1 + m 2 ^2 ^'^^^^ 

 be replaced by n's. We can in the first place choose p^ so that m2 = 

 P2 m22 + ^l, where o ^ m^ < ni2,2, and then, using this pj, choose 

 Pi so that nij = p^ mj,^ + p^ m,,! + m}, where o ^ mj < nij,! . Next 

 we insert these values for nij and m 2 in m.^ Q^ + m, fij = Pi 

 fli + P2 iïî + nij fii + m2 ^2- But mi fii + m, «2 a^^cl Pi fli + P2 ^2 

 are periods; therefore m\ Q,^ + m* Q,^ is a period and yet o ^ mj < 

 muando^mj < m22. This can only be the case if m\ = m^ = o. 

 This proves that J2i, fij is a primitive pair. 



In particular 



^1 = Pii Ô1 + P12 fii 

 ^2 = P21 fii + P22 fii 



In the normal case the determinant of the coefficients cannot vanish; 

 for if it did ^2/ ^1 would be real. 



There is no difficulty in completing this part of the discussion by 

 showing that the periods of the system are now all expressible in the form 



ni fij + n2 ^2- 



The special system of 'periods [ 12] 



If all the other periods of the system arise from Q, by multiplication 

 by m's, we might have among the m's the set h,l,\, ... .ad inf. There 

 would then be infinitely small periods. The simplest case that can 

 occur is when the periods are all of the form p S2 + p^ ft\ where ûVîî 

 = l/n. The function f (u) with the periods S2, fiS though apparently 

 doubly periodic, is really simply periodic with one period = an aliquot 

 part of Î2, and all other periods are integral multiples of this. 



If all the other periods of the system arise from 12 l)y multiplications 

 by m's, where some of the m's are irrational, the complexity is still 

 further increased. It is usual to limit the generality by selecting a system 

 in which a second period 12' has to 12 an irrational ratio, while all the 

 other periods are of the form p 12 + q 12'. We have here a kind of 

 multiple periodicit}^, since there is no one period of which all the others 

 are multiples. 



Functions f (u) with infinitely small periods. The discovery by 

 Abel and Jacobi of the property of double periodicity for the elliptic 

 functions suggested at once the dcsirabiiitv of a througli examination 



