84 NEWTON S PRINCIPIA. 



Q d x x 2 d z 



V 



x &amp;gt; a (twice 



_ 



the sector I C K)= /~ ~T) 2 A g am a az 



O 7 Q 



99 j j x 2 d z a 2 a 1 



:: a 2 : x 2 ; and adz x __ = _ x 



^~V 1 



= twice the sector Y C X. 



^ 



Hence results this construction. Describe the curve a b Z, 



n 

 such that (D b = u) its equation shall be u = 



and the curve a c x such that (D c=&amp;lt;p) its equation may 

 be &amp;lt;p = /&quot;&quot; T~ = 13 2 ~- Tnen tne differentials of 



the areas of these curves, or udx and &amp;lt;p d x, being respec- 



\c^ d x v tt ax 



tively &quot;~A. ^ _ Q* and // ,7 _^I and 



^u \/ ^* V *&quot; J? 



, x 2 d z ^ adz , 

 those being equal to and &amp;gt; or the sectors which 



are the differentials of the areas VIC and V X C, the areas 

 themselves are equal to those areas ; and therefore from 

 V X C being given (if the area c D V a be found), and 

 the radius C V being given in position and magnitude, the 

 angle V C X is given ; and from C X being given in 

 position, and C V in magnitude and position, and also 

 the area CIV, (if V D b a be found), the point I is 

 found, and the curve V I K is known. This, however, 

 depends upon the quantities made equal to u and &amp;lt;p 



