of the conic Sections. 
4 % 
But, as the transformation now mentioned requires but a mo- 
derate skill in series, I shall, for the sake of brevity, omit 
examples of it, and proceed to 
Sect. III. Examples of the Use of the foregoing Theorems. 
23. My intention in this section is, to illustrate the use of the 
foregoing theorems by a few examples, selecting at the same 
time such of the theorems as are best adapted to the case in 
hand; by which, and attention to what was said in the first 
section, of the limits of the convergency of the several series, I 
hope the reader will be directed how to make a proper choice 
of theorems on all other occasions. 
Example i. 
Let there be an hyperbola of which the semi-axes are 40 and 
30 respectively, and the ordinate is 10 ; it is required to find the 
length of the arch from the vertex of the ordinate. 
Since the conjugate semi-axis of this hyperbola is 30, we 
must, in order to fit the given numbers to our theorems, divide 
them all by 30 ; and then we shall have the corresponding di- 
mensions of a similar hyperbola as follows ; viz. the transverse 
semi-axis == ■§-, the conjugate semi-axis = 1, and the ordinate 
= And the proper theorem to be used in this case is the 
second. 
Writing, then, f for a, and $ for y, in the lid Theorem, we 
have 
ee 
aa -j- 1 = — , and 
3P 
MDCCCII. 
