Hoyal Astronomical Society. 141 



approximative process, and from them the corresponding values of ^, 

 Y), ^ are deduced. For all the subsequent values of ^, rj, 'C, the com- 

 putation is uniform. The mode of procedure is this : — From the 



three last values of — ^ an approximate value of the same quantity for 



the following argument is readily deduced, and a provisional value of ^ 



is then found by means of the formula of integration ^ = 'y-f ^ — , 



'y being a numerical quantity computed with the utmost facility by 

 the aid of results already obtained. The corresponding values of r] 

 and 'C being found by a similar process, the three resulting numbers 

 are employed in computing the second part of the expression under 

 the integral sign in each of the fundamental equations. Finally, the 

 numbers thus found, when applied to X, Y, Z, give the corrected 



values of — — ?, ~, which in their turn lead by double summation 

 df dt^ de ^ 



to the true values of };, r], I. By a similar process the values of ^, rj, ^ 

 for the next argument are ascertained, and so on in succession, the 

 operation being rapidly carried on throughout the entire period for 

 which the perturbations are to be computed. 



In order to test the practical utility of this method, M. Encke com- 

 pared its results with those obtained by the method of the variation 

 of arbitrary constants. For this purpose the perturbations of Vesta 

 by Jupiter from 1854, May 24, to 1855, October 7, were subjected 

 to a double computation. The following table exhibits the values of 

 the polar coordinates of the disturbed planet at the end of this period, 

 as resulting from the use of both methods : — 



Variation of Constants. New IMethod. 



Longitude 335 20 6-4 335 20 g'-B 



Latitude -5 40 42-3 -5 40 42-3 



Log. Had. Vector ... 0-3665032 0-3665042 



M. Encke mentions as aproof of the easy application of this method, 

 that in computing the perturbations of Vesta with given places of 

 Jupiter for thirteen epochs, only eleven hours were occupied, not- 

 withstanding that he was frequently interrupted in the course of the 

 operation. 



If extreme accuracy is required, the values of x, y, z, resulting 

 from the application of ^, t], ^, to x°, y°, z°, may be employed in 

 computing X, Y, Z, and then by repeating the operation we may 

 obtain corrected values of 1,7], C, This process, which gives the 

 perturbations depending on the square of the disturbing force, may 

 be repeated over and over again, until the assumed and resulting 

 values oi ^, r), ^ finally agree. The method, however, is so rapidly 

 exhaustive that this object is generally secured at the first repetition. 



When the perturbations are large the method is still practicable, 

 but it becomes necessary in this case to reduce in a corresponding 

 degree the magnitude of the intervals of quadrature. 



The following are some of the principal advantages attending this 

 method of computation : — 

 ♦ 1. It proceeds from the fundamental equations of motion by a 



