176 Transactions of the JRoyal Society of South Africa. 
The law of formation of the continued products is thus evident, and we can 
with confidence affirm the next result to be 
(a^ + ao^u + a-^a"^ + a^oj'^) ( ) ( ) + (^2" + <^3^^ + d^u^) 
^1 ^2 ^3 ^* 
\ &4 ^3 h 
h h h h 
h h \ h 
\ h h \ 
^4 ^1 ^-2 ^3 
C3 Cj. C]^ C3 
Cg C3 
^1 ^4 ^3 ^2 
(^2 
C^o 
^4 ^3 ^^2 ^1 1 
(7) Using S before any one of the products thus dealt with to indicate 
that u is to be replaced by w^, u'^, w*, and the results to be added to the 
original, we obtain from §5 
2(^1 + a^o) + -f a^to^)(bi + + ^3*"" + ^4^^) 
a-^ ttc) (X<^ 6?'^ 
" '^1 +1+ 1 + 1 
w -|- w ' -f- a>3 -f - 1 
w3+ 1 +0.2+1 
UJ^ + + w +1 
^1 ^4 ^3 ^2 
63 61 ?>3 
63 62 \ \ 
h h \ h 
4. 
Ctc) Oj>^ (X^ 
^1 ^4 ^3 ^2 ' 
because of the vanishing of 1 + w + + 
there are gfot 
n-1 
Similarly from §6 
^ Ol (Xo O3 O^ 
' ^1 ^4 ^3 ^2 ^1 
?>2 61 &4 63 
\ 63 &i 64 
^4 ^3 h \ 
Co* 
and S(ai + aoW + a3w3_^a4a>^)( )( ) (^/^ + C^o^ + C?3a>2 + £^^a>3) 
^ = 4 
^Sf-^ (Xt^ ^4 
^1 ^4 ^3 ^2 
h \ h h 
h h \ h 
^1 ^2 ^3 ^4 
^4 '^l "^2 "^3 
C3 C2 
^2 ^3 ^4 ^1 
^4 ^3 h ^1 
C^2^ i^t^ do 
The last is the main result, the others being derivable from it by speciali- 
sation. 
