30' PHILOSOPHICAL TRANSACTIONS. [aNNO 1756. 



for many ages the obliquity of the ecliptic has always decreased. But to show 

 this particularly: the secular motion of Jupiter's node, by prop. 2, is 10' 22-f ''; 

 and the annual motion of the equinox being 5(y', its motion in the same time is 

 1" 23' 20''; therefore the difference of the motions of the node and equinox, is 

 to the motion of the node, as 7.0331 to 1; therefore the time of the node's 

 transit from the vernal equinox to the autumnal, which gives the termination of 

 the diminution of the ecliptic's obliquity, will be in 14803 years, independent 

 of the small acceleration due to the force of Saturn. Jupiter's node then being 

 at present in S^° of Cancer, it appears that for 8000 years (if such be supposed 

 the age of the world) the obliquity of the ecliptic has decreased, and that it will 

 continue to decrease for more than 6000 years longer, and will not recover its 

 first situation till after a period of 29606 years. But the total diminution, which 

 in the aforesaid time can be generated in the obliquity of the ecliptic by the force 

 of Jupiter, produces by the theorem in this prop. 22' 30"; and therefore this is 

 the maximum of the variation. 



If there be required the decrease of the ecliptic's obliquity in the space of the 

 last 1000 years, it will be thus easily computed. The motion of Jupiter's node, 

 by prop. 2, in 1000 years is I'' 43' 44''; and the precession of the equinoxes in 

 the same time is 13° 53' 20", the difference of which motions is 12° 9' 36"; 

 hence, supposing the place of the node in the beginning of the year 1755 to 

 be in 8° 20' of Cancer, according to Dr. Halley's astronomical tables, the dis- 

 tances of the node from the equinox at the beginning and end of the given time 

 would be 93° 49' ZQ" and 81° 40'; and hence, by the foregoing theorem is 

 obtained 2' 22'' 56'" for the required decrement by the force of Jupiter. In like 

 manner, the motion of Saturn's node, by prop. 2, iR 1000 years, is 5' 56-^"; 

 hence the difference between the motion of the node and that of the equinox, 

 is to the motion of the node, as 1 39.265 to 1 ; but the distances of the node 

 from the equinox, at the beginning and end of the given time, according to 

 this ratio would be 68° 38' 24" and 82° 25' 48", supposing the node according 

 to the same tables to be in 21° 21' 36" of Cancer, in the beginning of the year 

 1755; and hence, the inclination of Saturn's orbit to the ecliptic being 2° SC 

 10", by the same theorem the decrement produced by the force of Saturn, comes 

 out 15" 1'", Therefore the whole decrement of the ecliptic's obliquity in the 

 last 1000 years, by the united forces of Jupiter and Saturn, becomes 2' 38". 

 So that from the time of Hipparchus the diminution of the obliquity of the 

 ecliptic is about 5'. 



Thus also, if Jupiter's ascending node in the beginning of the year 1750, be 

 in 8° 15' 50" of Cancer, and Saturn's node in 21° 20' 6" of Cancer, as in 

 Halley's tables, the following little table will come out: 



