1306 



Multiplying the second and third of (30) by /?, and then substi- 

 tuting in them the values of ^PiE and i^QiE derived from the first 

 and last we find equations which contain only P'/E and Q'iE- 

 These have the form 



(i?' ~E') Q'i, Ê + - - Gi, J, E, F Q'j.F + ^ 2 ^/, J, E, F P'j,F- 

 j F j F 



i^^-E^) P'i,E+ ^ 2: (r,j,E,FQ'j,F + ^^fPij, efP'j,f,\ 



J F j F 



(33) 



the coefticients G,H,G',H' being all at. least of the second order. 

 We have 



Pi^ = <'•'"> P'-f 4,0 ^= c'. 1 Qi, = c'„ ^'i-|-4,0 = cj.' 



The infinite determinant resulting from the elimination of P' and 

 Q' from (33) has, for each argument E, 16 rows and columns, 

 corresponding to the 16 unknowns P'ie and Qij,:. For ^^=0 there 

 are only 8 unknowns, and also the first of (33) becomes an identity 

 for E=: 0, so that there are only 8 columns and rows. The deter- 

 minant formed by the elements common to these 8 columns and 

 rows may be called the central determinant. 



All elements of the determinant outside the diagonal are of the 

 second oider '). The elements of the diagonal have the form 

 (j^ -i^ E- — /?*, where G is of the second order at least. In the 

 central square we have E=i), outside the central square E has a 

 finite value, and therefore G -f" E* is of the order zero. The manner 

 in which the determinant is reduced to its central square will be 

 explained by a simple example, in which I take for each argument 

 E only 2 instead of 16 rows and columns, and of the rest of the 

 determinant also only 2 rows and columns are written. This is 

 suflicient to illustrate the principle. We then have the transformation 



«11— Pf' «H «18 



«,i «11— i3* «,i 



') This is not correct. There are elements outside the diagonal of the orders 

 zero and one. The conclusions reached in the text are however not affected and 

 remain correct. For a more thorough discussion see Leiden Annals XII. 1. (Note 

 added in the English translation). 



