46 NOTE ON THE TRANSFORMATION OF A TRIGONOMETRICAL EXPRESSION. [110 
D_ 1, (a —c)(l + | 2 ), + ^ 
ft, ft+ft ft + i 
(aj-c) 2 
fVft 
O - c ) § 
fVft 
_ (a-cyf 
ftW 1 
ft, ft ft + i 
1, ft r 
- ft?? 
1, ft ft 
(a-^+^+r-ft??) 
ft ft ft 
or, replacing £, 77, £ by their values, we have identically 
1, x, {a + x)\/(c + x) 
ft V, (« + £/) V(c + y) 
1, 2, (a + ft) V(c + 2) 
(c+;ft)"(c+y)^(c+ft)^ j /a—c /cl—c ja—c ja—c ja—c / 
(a — c) 2 (V c+# v c+y + V c+£ V c+#v o+«V 
'a—c 
c+y'V c+2 
V- 
a — c 
a—c 
c+x’ 
« ! 
+ 1 
& 1 
1 a—c 
a—c 
c+y’ 
c+y 
'a—c 
a—c 
c+z' 
c+z 
and the equation 
/a — c /a — c fa —c ja — c ja — c j a — c _ q 
v c + # v c + y v c + .z v c + #v c + y V c + ^ 
is of course equivalent to the trigonometrical equation 
1-1 J + tan_1 J -£- C + tan- 1 a/ ^- C = 0, 
V c + a; V c + y V c + 2 
tan~ 
which shows the equivalence of the two equations in question.