1903-4.] Dr Muir on the Theory of Continuants. 
151 
Schlafli, L. (1858). 
[On the multiple integral J dxdy . . . dz whose limits are 
Pi = a Y x + \y + . . . 4- V > 0, p 2 > 0, . . . , p n > 0, 
and x 2 + y 2 4- . . . +z 2 > 1. Quart. Journ. of Math., ii. 
pp. 269-301, iii, pp. 54-68, 97-108.] 
The determinant which makes its appearance in the course of 
Schlafli’s research is 
1 
- COS a 
- cos a 
1 
- cos ft . 
- COS /? 
1 
1 
- COS 7] 
. . . — COS 7 ] 
1 
- cos # 
. 
- cos# 
1 
which for shortness’ sake he denotes by 
A ( a ,Ay> • • • ,v>0) 
and whose connection with continued fractions he therefore 
specifies by the equation 
A(q,/3,y, ■ ■ . _ , _ eos 2 a 
A(/3,y, . . . ,v,6) 1 - —j— _ „ 
cosb? 
1 - cos 2 # 
The first property noticed is, naturally, 
A («,Ay, • • • >0) = A (Ay, • • - , 0 ) - cos 2 a-A(y, . . . ,#). 
Later there is given what may be viewed as an extension of this, 
viz., 
A(a, . . . ,8,e,£,r),6, . . . ,\) = A(a, . . . ,S,c) • A (r/,0, . . . ,X) 
- cos 2 £ • A(a, . . .,8)-A(0, . . .,A), 
the proof being said to present no difficulty. The third is a little 
more complicated, and is logically led up to by taking four instances 
of the first property, viz., 
A (a,/3,y, • • • ,£) = A (Ay, • • • ,£) - cos 2 a • A(y,8, . . . ,£), 
A (fty,8 • • • ,C,y) = A (yA . . . ,rj) - cos 2 /? • A(8, . . . 
A (y, 8 . . • = A(y,8, . . . ,7]) - cos 2 # • A(y,8, . . . ,f), 
A (S, • • = A (S, . . . ,£,y,0) - cos 2 a- A(S, . . . 
