on the determination of the Orbits of Comets. 75 



Among the various changes to which the comet's apparent path 

 is subject, and of which an arbitrary choice may be made, for the 

 purpose of determining the distance in the cases in which the 

 general equation fails, or becomes unsafe, the author considers the 

 following to be the best : — viz. first, the curvature of the comet's 

 path, produced by the sun's action (or the deflection measured only 

 in the direction per-pendicular to the apparent path) ; second, the 

 acceleration in its path, produced by the sun's action (or the deflec- 

 tion measured only in the direction of its path) ; third, the deflection 

 in the direction in which both the sun's action on the comet and the 

 sun's action on the earth would cause a change of the comet's appa- 

 rent place (or the deflection measured along the gi'eat circle joining 

 the comet with the sun). These changes are severally considered, 

 and the method of forming the equation proper for each condition 

 explained, and rules deduced for the guidance of the computer in all 

 the particular cases in which the direct method cannot be followed. 

 In these investigations the correction of observed places of the comet 

 for parallax is entirely omitted, as it is most convenient, when p is ap- 

 proximately found, to correct the observations for the corresponding 

 parallax, to make the proper alteration in the second differential co- 

 efficients, and then to repeat the process of approximation to the 

 value of p. 



Having given the methods for finding the distance and its dif- 

 ferential coeflScient, the author concludes his first section with an 

 indication of the process by which the elements of the oi'bit are 

 computed. In the rules for the selection of the equations on the 

 parabolic assumption, some considerations are introduced which are 

 new and important. 



The second section contains remarks on the method of obtaining 

 numerical values of the differential coefficients of the right ascen- 

 sion and declination from the observations. In the use of these 

 quantities, what we have to consider is, not the effect of absolute 

 error in their values, but of proportional error. An error of a single 

 second in the value of the second differential coefficient of M. may 

 produce an ultimate error as great as would be produced by twenty 

 seconds in the value of the first differential coeflficient ; or as great 

 as would be produced by ten minutes in the M, itself. This con- 

 sideration allows the computer to determine many of the numbers 

 which enter into the equations after the second observation : the 

 method of proceeding is as follows : — 



" Adopt for the epoch the middle time between the first and 

 second observations : then the first differential coefficients of a and 

 /3 (a denoting the right ascension, /3 the declination) will be obtained 

 accurately by dividing the changes of a and /3 by the intervening 

 time ; and the values of a and /3 for the epoch will be obtained 

 with sufficient accuracy, by taking the means of a and /3 for the two 

 observations." 



The third and last section of the Memoir gives practical rules 

 for the computation of the observations. The successive steps of 

 the process, from the first observations to the determination of the 



