650 Proceedings of Royal Society of Edinburgh. [sess. 
. -1 . 
. -1 . 
+ oq 
1 . - 1 
. 1 . 
+ a 2 
. . -1 
. 1 . 
+ a 3 
1 
+ 1 
-1 
1 
and thence ultimately 
oq a 2 a 3 a 4 4- a 1 a 2 + a Y a 4 + <x 3 a 4 + 1 . 
Apparently as a consequence of this there is then formulated the 
familiar ‘rule’ referred to by Sylvester in his first paper of 1853. 
The other matters dealt with do not call for repetition. 
Hattendorff, K. (1872, March). 
[Einleitung in die Lehre von den Determinanten. xii + 60 pp. 
Hannover.] 
This is an introductory text-hook for school use ; and it is 
merely of passing interest to note that it contains a paragraph 
(§ 23) in which the fundamental relation between determinants 
and continued fractions is briefly but accurately stated and verified. 
Casorati, F. (1872). 
[Le proprieta cardinali degli strumenti ottici anche non centrati, 
Rendiconti del Reale Istituto Lombardo , v., fasc. 
iv., 13 pp.] 
In the course of his investigation Casorati has to deal with a 
series of quantities /3 0 , /3 1 , /3 2 , . . . . , each of which, after the 
second, is dependent on the two preceding it in the manner indi- 
cated by the set of equations 
P 2 ~ u iPi F P 0 
P 3 = U 2^2 F Pi F 7r i 
Pa ~ u 3^s F fi 2 
Pb : u aPa + As + 77 3 
Pe ~ u oPb F P 4 
P 7 = U &f > 6 F Ps + ^ 
Ps ~ Uy iPi + Pq 
