152 
Proceedings of Royal Society of Edinburgh . [sess. 
using in connection with these the multipliers 
A(S, • • . £,rj) , - A(8, . A (8, • • • ,£) , - A(y,8, . . . ,£) , 
respectively, performing addition, and then showing that the right- 
hand sum vanishes, the result thus being 
A(a,/3,y,S, . . . ,£) • A(8, . . . f,rj) - A(8, . . . f,rj,0, a ) • A(y,8, ...,£) 
= {A(fty,8, . . . ,£,?;) - A(y,8, . . . f,r),6)} • A(8, . . . ,£) • 
The fourth property concerns the determinant 
A(/?,y, • . • , 17 , 0 ) A(a,/J,y, . . . , 77 , (9) 
A(fty, . . . , 1 ;) A(a,/3,y, . . . , 77 ) 
which by reason of the first property can he shown equal to 
A (Ay, . • . ,?7,0) 
A(Ay, . • • ,v) 
~ A(y, . . . ,77,0) 
- A(y, ... ,77) 
COS^a, 
or 
A(y, • • • ,rj,0) A(Ay, . . . ,y,0) 
A(y, . . . , 77 ) A(Ay, • • • ,v) 
COS z a, 
and ultimately, “by repeating this sort of transformation,” equal 
to 
cos 2 a cos 2 /3 cos 2 y .... cos 2 0 . 
If we use for a moment the present-day notation for continuants, 
viz., where 
+ A + 
a 0 + 
a.^ + 
CL-^ Cfjcy 
h 
^2 ^3 
Schlafli’s results are 
seen 
to be 
k (fff* ) 
1 = 
k(& 
) + 
Pi ft 2 P< 
\l 1 ... 1 1 . . 
Pn 
. 1 
) = K ( 
Pi P 2 
1 1 .. 
- 
A k 
(A. 
P *- 2 
1 
\ . k( ^ k + 2 
>7 U 1 . . . 
