364 



Mr. T. Muir on an Extension of 



Now the fraction F remains unchanged when the sign of a 

 and the signs of all the &'s are changed ; hence, making this 

 substitution, there results 



£-* , , h(h-*) 1/1 v a -/3 F 



P b * + b ± + .. &,(&,-«) 

 '+ £-&„ + * 

 and therefore by multiplication we have the remarkable result 



+ h 



F 



+ a , b 2 (b 2 — u) , /z v 



•g~"- 5 * + a i A ft i ^( ft 3-«) 



g=!! + 8, + Afe-») 



/3-^2 + ^3 + 



c 3 (6 2 — a) 

 i3-b 3 + b± + 



, b n (b n —u) 

 P + b n -a 



+ 



s 



b „(b n — a. ) 

 fi-b n + a 



(C) 



where it is easily seen in what cases the fractions may be con- 

 tinued ad infinitum. 



Putting a=0 in this, we have Professor Bauer's * first case ; 

 and his second case is obtained on putting <x = b — c, and 

 b 2 , b 3l b±. . ,=b, b + h,b + 2 A, . . . respectively. Further, if after 

 these last substitutions we write 2s + h for /3, and h—b for c, 

 there results Euler's less general theorem f ; and if from this 

 we specialize still further by taking s = a — 1, h = 2, and 6 = 1, 

 we find the old well-known identity of WallisJ, used in the 



establishment of Brounker's expression for 

 ,xU 1 & 



5 2 



* • ' > a + JL)+ — 

 x 



7T 



VIZ. 



a— 1 + 



2(a + l) + 

 1 



2(a-l) + 



2(a + I) + 



3 2 

 2(a-l) + 



2(a + l)+. 

 5' 2 



2(a-l) + , 



Returning to (B) and taking b 2 from both sides, we have 



* Von einem Kettenbruche Enter's u. s. w. Miinchen, 1872. 

 t Comment. Acad. Petropol. vol. xi. 1739, p. 57, § 46. 

 \ AritJtmetica Infinitorum, Prop. CXCI. 



