of the conic Sections « 457 
°f —77 -przTT is V (uu + aa) ; and, if the fluents of — 
^(uu + aa) v v 1 J ■> 3 uy/(uu + aa) 3 
•* • 
W(™+««) '» &c - are denoted b y A > B > c > &c - we 
V (wM-J-aa) — a 
shall have 
A— 
B 
C 
D 
a 
H. L. 
— -y/ (uu-{-aa) 
u 
A 
2 aa uu 
zaa 3 
— \J (ww-f aa ) 
3 B 
4 aa m 4 
4 aa 3 
— \Z(uu+aa) 
5 C 
6 aa u° 
6 aa 3 
&C. &C. 
And, by multiplying these quantities by their proper coefficients,, 
and collecting the products together, we shall have (Theorem 
VI,) 
f V (UU + aa) + i A + B + -JJg . C + D, &c. 
2 
2.4.6. 8 
\ — d. 
Here also, the terms L, ' 1 , &c . 
which are component parts of this series, always decrease while 
y increases from o ad infinitum; and therefore the length of any 
portion whatever of the hyperbola may be computed by this 
series also, when the value of the constant quantity d , to be 
taken from it, is known. But the case to which this theorem 
ought to be applied is, when y is equal to, or greater than 1. 
And it has an advantage over some of the descending series, in 
that the terms B, C, &c. are divided by aa, as will 
appear in the use of it. 
g. When a = 1, that is, when the hyperbola is equilateral, the 
fluxionary equation in Article 7 becomes z 
u uu 
y x V(kw + i) 
I 
