1904-5.] Dr Muir on the Theory of Continuants. 
and the fact that 
671 
sin nO = a sin (n - 1)0 — sin {n — 2)0 ; 
in other words that K n and sin nO satisfy the same difference- 
equation.* 
Gunther, S. (1874, Nov.). 
[Lehrbuch der Determinanten-Theorie fiir Studirende. viii + 
236 pp. Erlangen.] 
In view of the attention which Gunther had previously given to 
the subject, it was natural that when he came to write his text-book 
on Determinants he should assign space to the consideration of the 
special form connected with continued fractions. We are not sur- 
prised, therefore, to find a whole chapter (Kap. vi.) with the head- 
ing “ Kettenbruchdeterminanten.” It extends to 31 pages, and is 
mainly a clear and detailed reproduction of his own and other 
previous work. In the paragraph dealing with the early history 
of the functions Sylvester is not now referred to. 
Gunther, S. (1875, Oct.). 
[Das independente Bildungsgesetz der Kettenbriiche. Denksehr. 
d. k. Akad. d. Wiss. in Wien : Math.-Nat. CL xxxvi. pp. 
187-194. 
Of this paper the portion which is not introductory and semi- 
historical concerns the development of 
X — cq 
cq X — a 9 
a 2 x 
X 
- <* n -l 
a n— 1 
X 
or A say, 
in a series arranged according to powers of x, this being considered 
* In connection herewith it may be noted that corresponding problems in 
the theory of continued fractions date much earlier : for example, problem 
No. 40 of Crelle’s Journal , ii. (1827), p. 193, a solution of which is given by 
Th, Clausen in a paper with the title “Die Function — — - 
a + a -f- a -(- . . . 
durch die Anzahl der a ausgedriickt,” published in Crelles Journal , iii. pp. 
87-88. In our analysis of Ramus’ paper of 1856 the first of the two values 
there given for the continued fraction is that obtained by Clausen. 
