933 



hence : 



—b"k{bk-b,)a -(» -l)(67f (« -•ck")={n-l){h'/crxk-' - ^nb'k .oic"+' 4-(n+ l).n-"+-^ 

 As according to (24) — />"^- (bk—O,) = {bk—b,y 'l—b'k) •■ bk vk, i. e. 

 =:b'k(l — b'k), we have also: 



lb'k-{b'ky\ a = {n-\) {b' k)' a - 27ib' k ^n"+' + (" + 1) 'Vk'^^^ 

 or 



Ji {b'kY a— b'k a — 2n b'k .^•yfc"+' + (n-^l) .vk''+- = 0. 

 After substitution of .i'/fc''+' for b',ka, and division by ./•a"+' , we get: 

 nb'k — 1 ~ 2u b'k 4- (n + l) .Vk = 0, 

 or 



(?? -f- 1) .rk — tib'k = 1, 

 from which 



l-'Vk 



(y) 



.vk—b'k 



On account of the value found for a, we can now write for (29a) 

 6 — 6„ \» a;^-« + l — &'yfc A'" 



or also 



bk — bj ,v/fc" + i —Vkxh"" 



1 N '^'^ — ^'^ 1 



bk—b^ J xk — è'/fc 



(30) 



in which 



^--^0 ^^---^o ,, {bk—b,y \—xk 



; Xk = ; b k=-- —7 — ; n = — , . (30a) 



v — b„ Vk— 1\ bkvk 'Vk — b k 



while from {a), {^) taken into consideration, follows : 



bk—b,Y ^ b'k .u-" + i 



:= 1 ; .»;„" = a = ---- . . . (30^>) 



bq—bj xic b'k 



11. It is easily seen that the found equation (30) fulfils all con- 

 ditions. In the first place we get properly 



'bf,-b,y_ _-^_ 



bk—bj , .vk-b'k ' 

 for y ^ oc (,f = 0), which is in harmony with (30''). Secondly the first 

 member = 1 for v^=vk {'I'^^vk), the second member {xk —b'k) •■ {.vk — b'k) 

 also becoming =1. Thirdly for b ==: h^^, v =: v„ the first member :^ 0, 



b'k a 

 and the numerator of the second member = ,ck , as .'Co"=rö! is 



Xk" 



put. But b'k a = ,i'/t"+' , hence this numerator is also = i). 



liy differentiation of (30), considering that there b'k stands for 



