652 
Proceedings of Royal Society of Edinburgh. [sess. 
where on the right, as desired, no /T s occur except the first 
two.* 
Immediately on this he formulates the general result, distin- 
guishing, of course, the case of /3. 2r from that of /3 2r + 1 - 
A more noteworthy fact, however, hut unnoticed by Casorati, is 
that a preferable form of result is obtainable by expressing the 
last determinant in terms of the 7r’s and their cofactors. Writing 
, w 2 , . . . , ufj for the determinant which is the cofactor of , 
and not confining ourselves to the special case where 7r 2 , 7r 4 , 7r 6 , . . . 
all vanish, we should then have 
/^8 “ ( U 1 > Uc 2 ? • ’ • > Ut l) fil + { u 2 j u 3 > • • • J U t) Po ~ ( U 3 5 u i j • • • 5 U l) 7r l 
(“4 > ' * * ’ 
(u 1 )TT b 
77 & 
and, further, the consideration of two cases, viz., where the suffix 
of /3 is even and where it is odd, would not then be necessary. 
Bauer, G. (1872). 
[Yon einem Kettenbruche Euler’s und einem Theorem von Wallis. 
Abhandl. d. k. bayer. Acad. d. Wiss. (Munchen) II. Cl. xi. (2), 
pp. 99-116.] 
The continued fraction referred to is 
n 
m + 
, WtI 
™+ 1 +^r 2 + 
and the theorem is that announced by Wallis in connection with 
the identity 
3 2 
2 + 5 2 
2 + 
* It may be well to note in passing that if 7 t 2 , 7r 4 , 7r 6 , . . . had occurred in 
the 3rd, 5th, 7th, ... of the set of equations, respectively, the solution would 
have been equally simple, the only difference then being that the first column 
of the last determinant would have had these quantities in the 2nd, 4th, 6th, 
. . . places. 
