.r/.^'^+i : /// , thus pi-operly the value given by f 30''). It is also easy to 

 see that in the limiting case 7 = \/^, i.e. bk = b^, the e(iuatioii (30) 

 passes into h =z const. ^=h^. For then //y[==0, and the second member 

 becomes constantly =: 1, i.e. /> constant = h/, = h^. 



12. It is selt-evident that the equation (30) is not the only one 

 that will satisfy the different conditions. Thus in the second member 

 of (29) e. g. we might have assumed v — h in the denominator 

 instead of v — Vo. A calculation quite analogous to that in § 10 would 

 then again have enabled us to find a set of values for n and a. 

 But then they would have been less simple than with v — i\. It is 

 easily seen that the result is obtained by substituting (1 -|-/>V.) : (1 —///,) 

 in ??, = (l—a?^.) : (.tVc— *'>'/.) for unity, and //a- : (1 —///•) for // M, so that 

 we then get: 



^k : xt — — ; .t'o = i/ a—xk 



having 



h—b.^lX ,vk—b'k:(\-b'f,) 

 The equation b=:/(v) itself becomes: 



b'k (x 



b — b.\" \—b't\.vi. 



bk-bj xk-b'k:{\-b\) 



But when we now wish to express n in 7 we get, instead of 

 n = 8y (7 -|- 1) : (27 — 1) (47 -|- 1) according to (31), the less satisfactoi-y 

 expression (IO7 + 1) : (27— 1) (47+1), in which IO7 + 1 = 5/>/, + b,) -. b^ 

 has a much less simple signification than 87 (7 -|- I) = bk rj- : b„'\ 

 We remind of the fact that a;k is now == (bt -- 60) : {vk — bk), hence 

 = (27 - 1) : 2. 



And there will be no doubt that more such functions are to be found, 

 which lead to more or less complicated expressions — but we confine 

 ourselves to the above. 



1) For where we found above ^7- = . «/,"—' X (f/j- — b'kxk) '■ {<-i — ^X"), a factor 

 1 — b'k will now occur by the side of xk~, so tliat after division of both members 

 by 1 — b'k everywhere b'k has been replaced by t'/j:(l — b'k). And in the equation 

 ïor b"k,[b'k — (b'k)~]a in the first member will be replaced by {ö'^—(6'/.)~) (a+.a" + '): 

 (1 — b'k)", when both members are now divided by {1 — h'k)~- On account of this 

 b'k: {I — b'k) comes everywhere in the second member, where b'k stood before, 

 while in the first member (//a— (/>'A-)^)X(a+.X(-"+'), is substituted for (ö'/c— (ö'/.)^) «, 

 the consequence of which is that, after application of entirely the same reductions 

 as above, ï^b'k — 1 — 2nb'k -\- ('i -f 1) Xk = is replaced by nb'k '• {\ — b'k) — 

 — (1 -t- b'k) : (1 — b'k) — 'inb'k : (1 — b'k ) + {n-\-\)xk = 0. 



