I 2^ ] 
that is, when e coincides with d (fig. 2.), and p with c 
(fig. I .), by what I have proved in the before mentioned 
paper, art. 10. 
4. From what is done above, the following ufeful 
theorems are deduced. 
THEOREM I, 
' 
The fluent of 4 <2 " ^ zx — 
H-z 
is = de. 
a — z 
THEOREM II. 
The fluent of 4 <2 " 5? 
T • a — z 
Z X — 
^=^+ilde-^ + ai.ef. 
+ Z1 
The fluent of 
THEOREM III. 
= 2 ef-de =2 F-E + AD-DP. 
-I r 
— z^U 
THEOREM IV. 
The fluent of M 
— ~ r:2xde-ef. N.B./^=! . 
^^ + 2/^z— z'^h 2a 
Thefe theorems flill refer to fig. i. 2. 3.; but now 
the values of the feveral lines therein (being not as be- 
fore) are as here fpecified ; videlicet^ 
Fig. I. In the hyperbola ad, the femi-tranfverfe axis 
AC is now = a ; the femi-conjugate = b ; the perpendicu- 
lar CP, from the center c upon the tangent dp, is = ^ 
the faid tangent dp x b"" + 2 b z- ; and the abfcifla 
tCB (correfponding to the ordinate bd) is =-^x 
_ai az + b" 
a^-\-b^ 
Fig. 
