228 Royal Astronomical Society. 



to do so affords an expression for the ratio of the distances at the 

 first and third ohservations on the usual assumption that the chord 

 is divided in the ratio of the times. 



" This expression may be converted into the elegant form given 

 by Olbers, so that it is identical with the value of M in his formulae, 

 and is expressed in terms that are likewise required in drawing the 

 line on the constant curve. 



" An example is given from the Trevandrum observations of the 

 great comet of 1843. The formulae are also applied to Gottinger's 

 observations of the second comet of 1813. A copy of the constant 

 curve is given upon a separate sheet, and the lines of these examples 

 drawn. The co-ordinates of the curve consist of the cotangent and 

 cube of the sine. It is easily constructed by the common tables. If 

 drawn with ordinary care, it will give the reading of the angle at the 

 eomet to greater nicety than even the best observations can afford. 



" I have appended a modification of Olbers's formula? for the radii 

 vectores and chord adapted to equatoreal positions, and involving the 

 use of the angular quantities already computed for the use of the con- 

 stant curve. The additional work of computation does not appear to 

 be so great as that which is required to convert the right ascension 

 and declination into latitude and longitude ; and, besides, it is easier 

 to compare observations with the computed elements when the latter 

 are referred to the equator. The inclination of the orbit and posi- 

 tion of the nodes are transferred to the ecliptic by the solution of one 

 spherical triangle. 



" In the recent improvements which Olbers has made in his me- 

 thod, by expanding Euler's formula into a series and reversing, the 

 means are afforded of constructing a small table which shortens con- 

 siderably the process of finding the distance by trial and error. An- 

 other improvement consists in the new expression given for the chord 

 being more favourable to accurate computation. I have included a 

 form of the same kind in terms of the right ascension and declina- 

 tion which is almost wholly made up of angular quantities that have 

 already been prepared and used with the constant curve. 



" In the last part of the paper an expression for the angle at the 

 comet is given to be used with the differential method which, in sol- 

 ving by trial and error, requires only five tabular references." 



Extract of a Letter from Sir John Herschel to the President, 

 dated Collingwood, November 29, 1845. 



"Being on the subject of the satellites of Saturn, I will mention 

 here a singularity which, though obvious enough, has not (so far as 

 I am aware) been noticed before, viz. that the periodic time of the 

 first satellite (first in order of the ring) is precisely half that of the 

 third, and the periodic time of the second precisely half that of the 

 fourth. This is far too remarkable and close a coincidence to be 

 merely casual, and (the second satellite being a certainty) the exten- 

 sion of the law to the first (a law so out of the way and unlikely) 

 would of itself be evidence of its real existence, even had it not been 

 (as it now certainly has been) re-observed. If such atoms perturb 

 one another's motions, there must be some very odd secular equations 



