1305 



The equations (30) can l)e satisfied by 



Xi = :^Ai, KVOS £t + 2: r MiJ,t BJ (03 {(fj + Et), I 



. • (31) 



y,- = 2; A' i, F sin ^t + S S l/'.-,j, e ^ sin {(fj -\- Et), ^ 



E j h 



where 



ffii = l^ir + '^iO' 

 Substituting these in (30) we find again equations of condition 

 for 1?^, Mi^^E a'lfl ^I'i,q,E- There is an infinite number of these equations. 

 Hence the condition for /J^ is an infinite determinant put equal to 

 zero. It is evident however that if 



is a loot, then all numbers of the form 



|i' = ± ^^ =b A: =h A;' c, {k, k' =z - CO . . . -\- <x) 



are also roots, since changing /? to ƒ does not affect x, and j, beyond 

 a change in the notation by which the different coefficients are 

 distinguished. 



It is not difficult to get an infinite determinant for ^^ instead 

 of ^. If we put 



Pij^ E = k {Mi J, E + Mij, _ e), 



P'i, j,E=ï {Mi, j, E - Mij^ _ Eh 



Q'iJ,E=HM.j,E+M<,j,_Eh 



Qi, j.E=k {M'.j, E - M'ij, - e), 



Then the equations become 



^Pi,E + EFi,E + è S £{ {gi,j,F-E + giJ.F+E) Q'j,F - 

 J F 

 — ifi. j. F-E + ƒ•, j, F-\-E) P'j,f\ = , 



^P'i, E-^EPi,E-{- h^^\ (9i.j, F-E — 9i.j,F+E) Qj,F - 



— {fi, j, F-E — Aj.I'-\-E) Pj.F I == 0, 



m'iE + EQi,E-\- è 2 ^;| {9' i. J, F-E + 9'i,j.F+E) Qj,F + 



+ {f'ij, F-E + f'ij. F+e) Pj,F \ = 0, 



mi E + EQ'i,E+ h 2 ^\{9'i,J,F-E - 9'i.j.F+E) Q'j,F + 

 j F 



+ if'i.J, F-E -y 'ij. F+e) P'j.F } = 0. 



(32) 



/ 



where we have omitted the index q in ^g, Pi.q.E, P'i,g,E etc. It is 

 only necessary to consider these equations for positive values of E. 

 The sums however include all values of F. We have 



Pi, F = Pi, -F, Qi, F^- Qi, -F, 



P'i, F=- P'i, -F, Q'i. F = Q'i, -F 



