450 PHILOSOPHICAL TRANSACTIONS. [aNNO 1773. 



sary to begin 29™ or 30™ before. It is essential not to begin too late, for fear 

 of missing the observation; it is also essential not to begin too soon, because 

 then the segment measured would be too small, as the satellite is continually 

 either approaching to Jupiter, or receding from him. All this is hastily ex- 

 plained ; but these matters are so familiar to you, that you cannot but under- 

 stand me, and this letter is already too long. I am afraid it will tire you; but I 

 am extremely desirous of having the exactness and utility of these two methods, 

 the one for the measure of the diameters, the other, for making all the obser- 

 vations capable of mutual comparison, ascertained, by repeating the observation 

 of the diaphragm in every eclipse. I cannot take a better way than to consult 

 the several astronomers, who, like you, besides being deeply skilled in the 

 theory, are the most celebrated observers. If they will adopt this method, it 

 will be the best way of making it general, as others will follow it of course. I 

 have communicated it to Mr. Messier, who proposes making use of it. Mr. 

 Marakli, who is gone to his house near Nice, has tried the observation of the 

 diaphragm with success, with an achromatic telescope 3-^ feet long; and he 

 would already have made use of it, but that it is impracticable with the telescope 

 of 1 5 feet, which he uses for the eclipses of the satellites. I have written to him 

 that he may observe the eclipse with his usual telescope, and the diaphragm with 

 the achromatic; so that I make no doubt he will use this method as soon as 

 Jupiter shall have come out of the sun's rays. These, Sir, are the things on 

 which I wish to consult you, and have your advice. I shall be much flattered 

 by your comnmnicating this letter to the h. s., if you think itdesei-ves attention. 



Notes on the Foregoing Paper. By the Rev. Samuel Horsley. p. 213. 



^Rr (a — b) 

 (<) i- -. This formula is deduced from the following principles. 1st. That the motion 



of the satellite, in its orbit, is uniform, or at least may be considered as such, witliout sensible error, 

 in the present investigation. 2. That the time which the semidiameter takes to enter the shadow, in 

 any eclipse, is inversely as tlie whole time of the duration of the eclipse. 3. That the time which 

 any given part of the semidiameter takes to enter the shadow, is to the time which tlie whole semidia- 

 meter takes to enter, as tliat part to the whole. 



Now, let a and b denote the versed sines of tlie arcs Ad, ad (in the first figure) respectively, the 

 radius being unity. Let a denote the half-time of the duration of an eclipse, when Jupiter is in the 

 node of the satellite's orbit, r, tlie time which the semidiameter takes to enter the shadow in such 

 eclipses ; d, the whole duration of an eclipse, happening when Jupiter is at any given distance from 



2Rr 

 the node. Then will — j- express the tune which the semidiameter of the satellite will take to enter 



the shadow, in the eclipse whose duration is d (hy 2*, because d: 2r = r : 



*2Rr 2Kr . . 2Rr (a — b) 



- ). And, — being the time that the semidiameter takes to enter the shadow, -^ will 



be the time that tlie part b6 takes to enter, by 3'. 



It is to be observed, that, to compare two eclipses by this formula, it is necessary, that the planet 

 should have been at the same distance from the node of the satellite's orbit, at the commencement of 



