824 On Continued Fractions in Quaternions. 



we see that 



(88; «,+i+A(i-» -^-^> vTI^-; • ■ • ^^''> 



yiheijce a^, b^ are two real quaternions, namely, ^^ '^^' 

 or, as we may also write them, 



Mi BB ^^.- ^^^^^Hu^_k)+i-'j. J * ' :'. ^/^[^"^ 



■Jn tms manner I found it possible to eliminate ttie symbol h, or 

 to return from imaginary to real quaternions ; and so perceived 



Mkj U>0-\ \»^% i) AgSftill — ^; U2n+l+Z_____ ^i^n+i ^ ^ ^^g. 

 0^^ 10 8J001 I^e l-'i ^2n* 1— y A2„+i v.ni'^'ftWi «v> 



■BdtB^'df ifiise two last formulse agree in giving, as a limfiF/S«^^' 



io xiiim)^i.ixij !fscj±i = 4- = ^ = + /t ; . . '<;:^ <i^ ^'^^^?^i9() 



-noo gi fi rfo l"-i ^ J ' ■ ija '10 ^{mdj^J^ 



and therefore (as in art. 5), '^vv*, \fob'v b y^ baiomia 



, . >.<» f* 1;-' :i ^'.Nv-jYiuo V. tfiioq 



(iliFr)«o=V=-^+*(l-7')=«. • (30) 



whatever real quaternion may be assumed for Uq. This last 

 restriction becomes here necessary, from the generality of the 

 analysis employed : because, for the very reason that u', u" are 

 admitted as being at least symUolical (or imaginary) roots of the 

 eciuatipn^(15), therefore we must here say that 



ifMQ=M', then w^=w', u^=u'; . . . (31) 

 -ItflST in iSke manner, 



.mi>)hfi if Wo=tt", then w^=m", M^ =tt". . . . (32) 



■^^H" tBy the first of the two real quaternion equations (28), 



l!^,,have, lo uii.; u' 



rr. W2„-^=-2-^ + A2„B-»(l--»; . -i^i-faS) 



k 



but also, by the latter of the two values (27), 



B^^'d-^^ {(1 -^r'B.,}" - (-f^ • B*.)"' 



= {l+n(A-.)(«o-*)}-; . . . (34) 

 again, by the former of the same two values (27), 

 A2„-(A: + t){l + n(A:-i)(wo-A;)} 



= Aa^~(A: + i)+2n;(wo-A:) = Wo-A:; . (35) 



-3VII00 



