178 Astronomical and Nautical Collections. 



equation tang 9 = . ^^"! ^ ., Hence we may easily find 

 ^ ^ sm (A — a ) 



the position of the intersection of these two circles with respect 



tano: VI 



to the ecliptic ; for if we put cot a = ^^ng S, in (A" + ^-a') 



4- cot (A" + 9r— «'), we have a'— tt + <t for the longitude of 

 this point, which may be called c", and its latitude y" will be 

 such that tang y' = tang u sin er. 



[Note 5. In order to demonstrate the general proposition, 

 that the relative apparent places of two bodies describing right 



(E) .^,^(D) 



i%r ::::-iG) / 



lines with proportional velocities, will be found in a great circle, 

 we must consider, that if the line AB be in the plane of 

 the figure, and CD meet it in C, the great circle, to which 

 both AC and BD will be directed, will be found by drawing 

 AE parallel to BD, and determining the plane which passes 

 through AC and AE. Now, if we take, instead of BD, 

 any other position of the line of direction, as FG, di- 

 viding AB and CD in the same ratio, and draw AH parallel to 

 FG, it may be shown, that AH will be in the same plane with 

 AC and AE; for if AE = BD and AH = FG, the points C, 

 H, and E will be in the same right line, their heights above the 

 plane of the figure being in the same proportion, as their dis- 

 tances from C, since these heights are equal to the heights of 

 G and D, which are to each other as CG to CD : and since 

 GH is equal and parallel to AF, it will be to DE as CG to 



J 



