I write for the moment f = A / & c - j this gives 
NOTE ON THE TRANSFORMATION OF A TRIGONOMETRICAL 
EXPRESSION. 
[From the Cambridge and Dublin Mathematical Journal, vol. ix. (1854), pp. 61—62.] 
The differential equation 
dx dy dz 
- 1 U Q 
(a + x)J(c + x) (a + y)\/(c + y) (a+z)J(c + z) 
integrated so as to be satisfied when the variables are simultaneously infinite, gives 
by direct integration 
tan - 
a — c 
c + x. 
+ tan~ 
a — c 
c + yj 
+ tan' 
a — c 
c + z. 
= 0; 
and, by Abel’s theorem, 
1, x, (a + x) J(c + x) =0. 
L V> 0 + y) V(c + y) 
1, z, {a + z) V(c + z) 
To show d 'posteriori the equivalence of these two equations, I represent the deter 
minant by the symbol □, and expressing it in the form 
□ = 
1, a + x, (a + x) \/(c + x) I,