VOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. SQ 



the variation in the obliquity of the ecliptic, as the sum or difference of the 

 sines of the distances of the node from the equinox at the beginning and end of 

 a given time, is to the sum or difference of the cosines of these distances." 



And, because it is always, as lw is to mn, so is the cosine of the obliquity- of 

 the ecliptic, to the sine of the same obliquity, or as radius to the tangent of the 

 same obliquity, then will the sum of all the Ln in a given time, i. e. the varia- 

 tion of the equinoctial point according to longitude measured from a fixed point 

 in the plane dce, be to the variation of the same according to latitude, in the 

 same ratio, and therefore is given, a. e. i. 



CoROL. Hence it follows that the variation of the vernal equinox according 

 to latitude, computed from an immovable plane, is always made towards the 

 north, in the passage of the ascending node of Jupiter or Saturn from the sum- 

 mer solstice to the winter ; and towards the south when the same node transits 

 from the winter solstice to the summer. The contrary must be said of the au- 

 tumnal equinox. But the variation of the equinox according to longitude, 

 counted from a given place in that immovable plane, is in the former case made 

 contrary to, but in the latter according to the series of the signs ; that is, in 

 the former case the equinox recedes backward, but in the latter precedes 

 forward. 



If the prints d and h should be situated on different sides of the solstice, i. e. 

 if in the time proposed the node should transit through the sign cancer or Capri- 

 corn, the theorems in the prop, will give the difference of the contrary variations 

 of the equinoctial point ; and k the sum of these may be had is easily seen. 



Scholium. Since in the course of the last 1000 years Jupiter's ascending 

 node has possessed the sign Cancer ; and because the aforesaid variations have 

 not been made sensible in the whole of that time, we may inquire what they 

 have been in the 500 years counted backward, from the beginning of the year 

 1755 : in which case the difference in the motions of Jupiter's node and the 

 equinox, is by the scholium to the preceding prop. 6" A' AS" ; hence proceeding 

 as there for the rest, either theorem in that prop, produces, for the variation of 

 the vernal equinox in north latitude, 6" 37'^', and hence the variation in longitude 

 is = 15'' \A"\ by Jupiter's force. 



By adding, in the former case, 1" 1&^' for the force of Saturn, and 5" 3&" 

 in the latter, then the total variation of the equinoctial point, according to la- 

 titude, in the last elapsed 500 years, is = g" 3'^\ and the regression of the same 

 point 20' 50'"'. So that the variations of this kind are not sensible unless in a 

 long interval of time. 



Prop. v. To Investigate the Equations of the Terrestrial Errors. 



The greatest equations of the angular errors are directly as the forces and the 

 square of the times, and inversely as the diameters of the orbits ; and therefore 



