of the Conic Sections . 
45 1 
Sect. I. The Investigations of the several Series. 
1. That the following processes may not be incumbered with 
symbols, and that the rate of convergency of the series obtained 
therefrom may be the more obvious, let the transverse axis of 
any hyperbola be called 2 a, and the conjugate axis 2 ; (by 
which notation, any ratio that these two lines can possibly have 
to each other may be expressed;)* let the abscissa be called x, 
the corresponding ordinate to the axis, y , and the length of the 
curve from the vertex to the ordinate, %. Then, by the well- 
known property of the curve, we have sax -f- xx = aayy ; from 
which x is found = ay/ ( 1 +yy) — a, and x = ay - — and 
*=S(yy, + i*) =jV(i + ^)= ; Which 
equation, by writing ee for 1 4- aa, will become z 
2. Now, the fluent of the expression on the right-hand side 
of the last equation may be taken in different series, according 
as the numerator or denominator of it is converted into series,, 
and according as 1, eeyy, or yy, is made the leading term. By 
converting the numerator, y' (1 + ee yy )> into series, making 1 
the leading term, we get z = - x : 1 ~j_ e lH _j» 
3^ y 6 
,8 „S 
2.4.6 &c - and then - b y taking the fluents of 
y yy y y^ o 
Tc'i+'JP Vi T+~’ &c * and denotin g tbem b y A, B, C, &c, 
respectively, we shall have 
y 
* With respect to homogeneity, about which some have shown more scrupulosity 
than discernment, I shall add a few words in a subsequent part. 
