Sept ii, 1884] 



NA TURE 



plane of the ecliptic. Now our clock and all measurements of 

 time must depend upon the earth's rotation, the plane of which 

 always remains parallel to itself, and we have seen that our 

 start-point for geocentric and heliocentric longitude depended 

 upon the fact that at a certain point in its revolution the earth 

 passed through a node, and that the node at which the sun 

 with its apparent motion crossed the equator northward was 

 called the ascending node. In the diagram this is represented 

 by r in the upper figure, and the descending node is indicated 

 by a in the lower figure. It will be seen that if we have equal 

 intervals along the ecliptic the motion along the equator is repre- 

 sented by bases of successive triangles, of which the hypo- 

 thenuses lie along the ecliptic. Now the hypothenuse must be 



<- 



W 



Fig. 51. — Diagram showing how the sun's apparent motion along the ecliptic, 

 now parallel with the earth's equator (the central line of the figure)at the 

 summer ( °b ) and winter ( V? ) solstices, is represented by equal intervals 

 along the equator. 



greater than the base, so that we have at the ascending node 

 the motion of a body along the ecliptic represented only by the 

 base of a triangle of which the motion itself represents the 

 hypothenuse ; and the same thing happens in the opposite manner 

 at the descending node ; whereas if we take the other positions 

 shown in Fig, 51, for a short time at all events the motion will 

 be parallel, and motion along the ecliptic will be represented by 

 an equal amount along the equator. 



These then are the difficulties we have to face when we come 

 to fix our sun- time, first, the unequal velocity of the earth round 

 the sun ; and secondly, those variations which are brought abouj 



by the fact that the two motions of the earth — its axial rotation 

 and yearly revolution — take place in different planes. How 

 are these difficulties got over ? They are got over by pretending 

 a, sun, as a child would say. Astronomers pretend that there is 

 a sun moving along the equator, or, in other words, they pre- 

 tend that the earth's movement of revolution takes place in the 

 same plane as its movement of rotation. It is further imagined 

 that this imaginary sun travels at precisely that rate which it 

 would il the average of all its rates along the ecliptic during a 

 year were taken, so that we get something like this (see Fig. 52) ; 

 first of all we have the curve bbbb, which shows the variation 

 which would take place providing we only had to deal with the 

 obliquity of the ecliptic. Where that curve crosses the horizontal 

 line, we get at those moments (if we disregard the elliptic motion) 

 the same time shown by the mean sun as we should get if the true 

 sun had been taken ; it will be seen this occurs four times during 

 the year — on March 20, June 21, September 23, and December 22. 

 Then there is another curve, c c c c, which represents another 

 relation between the mean sun and the true sttn. Providing 

 that the two planes were coincident, and that the movement 

 of the earth under these conditions were exactly the same as 

 under the present conditions, namely, that she moved in an 

 ellipse and that the radius vector swept over equal areas in equal 

 times, then we should have the true and mean sun coincident on 

 December 31 and July 1 only. Then the algebraic mean of 

 these two curves, BBBB and c c c c, is taken, and we get as a 

 result the lower curve dddd, which is a compound of the two 

 other curves, and as the result it will be seen that where we got 

 the curve c, giving us a difference of nearly five minutes, and the 

 curve B, giving a difference of about nine minutes in the same 

 direction, we have a very great departure between the motions 

 of the real and mean suns. Above and below the datum line, 

 which is marked zero, we have 5, 10, and 15, which repre- 

 sent the difference in minutes at which the southings of our 

 real and fictitious suns really take place. Early in the month of 

 February we have a difference of very nearly fifteen minutes 

 between the two suns, and it is at this time of the year of course 

 that the sun dial is most in error. At other points where the 

 effect of curve B is to cause a great difference, the effect of 

 curve c will be to minimise that difference, and so in the 

 compound curve D the difference is very slight. About the 

 middle of June we get them together, then towards the end of 

 July we get another separation, and about November I we come 



52.— Dis 



t showing how the equation of time (curve dddd) results rom the combination of curve 

 obliquity of the ecliptic, and curve cccc representing the difference between the 1 



to another difference even greater than that in February. In 

 this way a correction has been introduced, which is known as 

 the "equation of time," and this added to the motion of the 

 true sun, or added to that of our imaginary sun, brings them 

 together, and by this means the mean sun is kept as nearly as 

 possible to the average position of the true sun throughout the 

 year. Another diagram (Fig. 53) will enable us to understand 

 some of the considerations which have brought this about. Let 

 F represent the position of the sun in one of the foci of the ellipse, 

 PtA, round which the earth is supposed to be travelling. Now 

 while we have the real radius vector going from p to c, with its 



unequal motion along the orbit, we have a fictitious radius vector 

 going with absolute constancy along the circle. We get what is 

 called the true anomaly in the angle P F c, and the mean anomaly in 

 r Fi', and the difference ev e is called the equation of the centre. 

 This equation helps us to determine those curves to which refer- 

 ence has been made, and the chief object in calling attention to 

 this diagram is to explain the meaning of the term anomalistic 

 year, which it will be necessary to introduce presently. It has 

 already been said that it is imperative, if we are to gain any 

 advantage from it, that real sun-time and apparent sun-time 

 should never be widely separated, because if so we might have 



