24 RELATIONS PERTAINING SIMPLY [BOOK. 1. 



In the same manner, by adding 1 to both sides, it becomes 



vii. 



By dividing VI. by VII. we should reproduce III. : the multiplication produces 

 VIII. r sin v =pcoian y tan F= I tani/&amp;gt; tan F 



= i jo cotan y (u -- ) = i b tan y (u -- ) . 

 From the combination of the equations II. V. are easily derived 

 IX. rcosv=b(e -j,) = tb(2 



u 



22. 



By the differentiation of the formula IV. (regarding y as a constant quantity) 

 we get 



du , / , , . , x \ T rtanil; 



= i (tan 3 (v -4-w) tan * (v r 

 M \ i &amp;gt; ^~ 



hence, 



dpr , 

 n\ J. f\ ni 



or by substituting for r the value taken from X. 



MM u 



Afterwards by integrating in such a manner that the integral may vanish at the 

 perihelion, it becomes 



(}e(u ) \ogu}= 



The logarithm here is the hyperbolic; if we wish to use the logarithm from 

 Brigg s system, or in general from the system of which the modulus = \, and 



