58 PHILOSOPHICAL TRANSACTIONS. [ANNO 1781. 



ficient for extending the interval of time to those observations that were made in 

 the year 1777- But as these are the synodical revolutions, it will be necessary 

 first to reduce them to sydereal rotations. In fig. 7, pi. 1, let us suppose the 

 orbit of Mars, mabc, to be in the same plane with the orbit of the earth, edfg; 

 and the axis of Mars to be perpendicular to his orbit. Let m, e, m, e, be the 

 situations of Mars and the earth on the 13th of May and 17th of June ; then 

 will the line em, that connects the centres of Mars and the earth, point out the 

 geocentric place of Mars on the 13th of May ; and the line em, the geocentric 

 place of the same planet on the 17th of June. Draw er and ms parallel to er ; 

 then will er point out the geocentric place of Mars on the 13th of May ; and 

 the angle sme is equal to the angle mer. Now, by an ephemeris the geocentric 

 place of Mars, May 13, at ll h 20 m , was 7 s 20° 59' 21"; and on the 17th of 

 June, at g h Q m , it was 7 s 12° 27' 22", by which we obtain the difference or angle 

 rem = ems = 8° 31' 59". 



Now a spot on Mars, situated in the direction me, will have made a sydereal 

 revolution when it returns to the same, or a parallel direction ms. From which 

 we gather, that the spot on the 17th of June, after coming to the line me, 

 where it finishes the synodical revolution, will have to go through an arch of 

 8° 3l' 59", in order to arrive into the direction of the line ms, where it finishes 

 the sydereal rotation. The time it will take to go through this arch, at the syde- 

 real rate of 24 h 39 m 20 s to 360 degrees, or 4MO9 per minute of a degree, will 

 be 35 m 3\8 ; this being divided by the number of revolutions 34, gives l m l s -8 ; 

 which, added to 24 h 38 m 20 s .3, gives 24 h 3g m 22 .1 for the sydereal revolution 

 of Mars, as found by the 3d of the monthly periods. This quantity will help 

 us to find a proper divisor for the 3 following long biennial periods. 



It is to be observed, that Mars has been retrograde in the above example, for 

 which reason the measure of the angle ems was to be added to the synodical 

 revolution when we wanted to find the sydereal rotation ; but if he had been 

 direct, or if his place had been more advanced in the ecliptic than that to which 

 we compared it, as at p, then the line juct parallel to em would be the direction 

 to which the spot should return, in order to accomplish a sydereal revolution, 

 and therefore the quantity of the angle a^e = ^er, or difference of the geocen- 

 tric places ought to be subtracted from the synodical revolution to obtain the 

 sydereal one. 



First biennial period from 1777, April 8 d 7 h 30 m to 1779, June 6 d I0 h 10 = 

 789 d 2 h 40 m . 



The geocentric places of Mars at those times were, 6* 6° 3l' 26* and 7 s 13° 

 48' 30", their diff. I s 7° 17' &", turned into time, at 4MO9 per minute of a de- 

 gree, and subtracted, because Mars is more advanced in the ecliptic, is 78()'' 2 h 

 40- o 9 - 2 h 33 m 11\8 = 789 d h 6 m 48\2 = 768 revol. hence 1 revol. = 

 24 h 39 m 23'.03. 



