SECT. 3.] PLACES IN OKBIT. 133 



98. 



The solution of our problem for the ellipse in the preceding article, might be 

 rendered applicable also to the parabola and hyperbola, by considering the parab 

 ola as an ellipse, in which a and b would be infinite quantities, (f =. 90, finally 

 E, E , ff, and G = ; and in a like manner, the hyperbola as an ellipse, in which a 

 would be negative, and b,E,E ,g,Gr,(p, imaginary: we prefer, however, not to 

 employ these hypotheses, and to treat the problem for each of the conic sections 

 separately. In this way a remarkable analogy will readily show itself between 

 all three kinds. 



Retaining in the PARABOLA the symbols p, v, v , F,f, r, r , t with the same sig 

 nification with which they had been taken above, we have from the theory of the 

 parabolic motion : 



[2] y/ = **(* +/) 



2 Iff 



~ = tan * (,P+/) tan * (I 7 /) -f- i tan 3 i (I 7 +/) * tan 3 * (-F /) 

 (tan } (JP+/) tan * (Ff)) (l -f- tan } (^+/) tan i (^ /) + 

 i (tan * ( J H-/) tan * (I 7 /)) 2 ) 



2 siny^ / /2 cos y^ r / .4 sin 2 /V /\ 

 p \ p 3pp / 



whence 



rg-| T.4 _- 2sin/cog/.r/ . 4sin/(r/)^ 

 Further, by the multiplication of the equations 1, 2, is derived 



and by the addition of the squares, 



[5] 



