316 



the angular velocities of the moving system of axes about the three 

 axes themselves. In this expression w , ooj, ta. 2 may be any 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 coefficients 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 plane of 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 line of nodes, relatively to fixed space. 

 The mutual inclination of the two orbits being supposed known, 

 their several inclinations to the principal plane 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. 



IV. Extract of a Letter, dated January 6, 1855, from J. MIT- 

 CHELL, Esq., Quartermaster of Artillery, Bangalore, " On 

 the Influence of Local Altitude on the Burning of the Fuses 

 of Shells." 



" In the early part 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 being 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 



