T A N 



This will also give the time between two oppositions, or between any 

 two similar situations. 



= -It 



Cor. Since = -, p = 



Therefore, from the earth's period (P) known, and the synodic (*) ob- 

 served, we can determine the periodic time (/>) of the planet. 



For the synodical periods of the planets see Planets elements of. 



2. To find the same for three bodies. 



Let T = time between the conjunctions of the 1st and 2d found as 

 above j t = do. between the 2d and 3d. 



Then \fm = least common multiple of T and t, m time between two 

 .conjunctions of the three bodies. 



TANGENTS, method of drawing. 



1. Method of drawing a tangent to any curve, whose equation is gir 

 Let x and y abscissa and ordinate ; then 



Sub tangent . 



Ex. 1. In parabola, subtangent 2 or. 



y a x .r 

 a - 



2 a .v -f A- 

 3. In hyperbola, subtaji. . 



2. To find the equations to the tangent and normal. 



Lett/' a x' -f- & be the equation to the tung-cnt; then a -/T, 

 also since the curve and line have a point in common, // a ,r -\- I, 



and y' " y = -,-' (?' *') whifh is the requherl o<|uatiwu. 



Again let y" and x" be the coordinates to the normal, tiicu since it 

 passes through a point whose coordinates are ..r and //, and is perpoudL- 



