48 Royal Society \^- 



axes themselves. In this expression Wy, cy^ w^ '^^Y ^^ ^"X arbitrary 

 functions either of the time or of the elements; but in any case these 

 functions are to be exempted from differentiation with respect to the 

 elements in taking the partial differential coefKcients of the disturb- 

 ing function. 



This result is illustrated by referring the motion of a system con- 

 sisting of two planets to axes so chosen that the i:)lane qI xy shall 

 always coincide with the principal plane of the system, and the axis 

 of X, from which all longitudes are reckoned, shall always coincide 

 with the line of nodes ; there are thus obtained twelve rigorous 

 simultaneous differential equations, of which nine form a system 

 apart, containing only the major axes, excentricities, epochs, longi- 

 tudes of perihelia, and mutual inclination of the two orbits, and 

 afford an example of the so-called "elimination of the nodes;" 

 whilst the remaining three (which contain also these nine elements, 

 but not their differential coefficients) determine the motion of the 

 principal plane and of the Hne of nodes, relatively to fixed space. 

 The mutual inclination of the two orbits being supposed known, 

 their several inclinations to the principal j^lane are given by simple 

 relations ; and the positions of the planets in their orbits (their 

 longitudes reckoned from the line of nodes) being supposed known, 

 their motions relatively to fixed space would thus be completely 

 determined. 



Extract of a Letter, dated January 6, 1855, from J. Mitchell, 

 Esq., Quartermaster of Artillery, Bangalore, "On the Influence of 

 Local Altitude on the Burning of the Fuses of Shells." 



" In the early jjart of the year 1848, at the annual practice of the 

 Artillery in this garrison, it was observed that the fuses burned too 

 long a time. The regular burning of fuses beuig a matter of much 

 importance, the circumstance was duly reported to Artillery Head 

 Quarters, and a portion of each kind was directed to be sent to 

 St. Thomas's Mount (eight miles from Madras and on the same 

 level) for examination, where they were found to burn correctly, and 

 as at that time no one suspected the real cause of the discrepance, it 

 was concluded there had been some error in the length of our pen- 

 dulum, and there for the time the matter rested. 



" But as I was satisfied, for many reasons, that it was not owing to 

 an error in the pendulum, I determined to keep the subject in mind, 

 and for the practice in the following year (1849) I caused to be 

 made an adjustable pendulum which beat seconds very correctly for 

 several minutes. This pendulum was daily compared with two or 

 more seconds' watches to make all safe, and, as our longest time of 

 flight did not exceed twenty seconds, no error could possibly arise 

 from this source. The result was that the fuses again burnt too 

 long at Bangalore, and were again found to burn correctly at 

 St. Thomas's Mount. 



" This was a mystery to all; but after the matter had cost me much 

 thought, it occurred to me that the cause was to be sought in the 

 difference of altitude between St. Thomas's Mount and Bangalore, 

 nearly 3000 feet ; and as a means of putting this to the test, I sug- 



