1905 - 6 .] Dr Muir on the Theory of Alternants. 
369 
Cayley (1853). 
[Note on the transformation of a trigonometrical expression. 
Cambridge and Dublin Math. Journ., ix. pp. 61, 62 : or 
Collected Math. Papers , ii. pp. 45, 46.] 
In order to show that the vanishing of the alternating function 
1 
1 
1 
implies the vanishing of 
(a + x) fc + x 
(a + y) Jc + y 
(a + z) J c + z 
tan 
-i / °LAL + tan " 1 / — - + tan " 1 / 
V c + x V c + y v 
c + z 
c + x ^ c + y 
the determinant is proved to contain the factor 
fa-c la — c _ la — c ja — c J 
'y c+x V c + y c + z c + x\/ c + y V 
with the cofactor 
_ (c + x)%(c + y)i(c + z)% 
(«-«)’ 
a - c 
c + z 
j 
■j 
'a- c 
a - c 
c + x 
c + x 
'a - c 
a - c 
c + y 
c + y 
'a — c 
a - c 
c + z 
c + z 
'a -- c 
c + y 
j 
a - c 
c + z 
This is done by writing £ , y , £ for / - — - , A 
\ c + x V 
respectively,* and so changing the given determinant into 
* It is simpler still to express the given determinant in terms of alter- 
nants having \Jc + x , \/c + y , \/c + z f° r variables, 
determinant 
1 x {c + x + a - c)\/c + x 
1 y (c + y + a- c)\Jc + y 
1 z (c + z + a-c)\J c + z 
Thus the given 
= 
1 c + x {c + x)\Jc + x 
+ 
1 c + x > Jc -t- X 
1 c + y {c + y)\/c + y 
1 c + y \Jc + y 
1 c + z ( c + z)\Jc + z 
1 c + z \Jc+z 
= 
1 A A - (a-c) 1 
y v? j 
say, ' 
1 V v 2 
1 C C 2 
/j. A I • {yv + v£+{y.-(a-cj) • 
V v 2 
c H 
PROC. ROY. SOC. EDIN. — YOL. XXVI. 
24 
