b REPORT OF THE SECRETARY. 



the author was enabled to devise a method more simple than any before 

 employed, of which the quantities previously calculated furnished the 

 basis, and which has the great advantage of being easily applicable to 

 the perturbations of the second order. The results of the two computa- 

 tions have been found to agree very closely. The more important terms 

 of the second order have been once computed, but will be gone over 

 agaiu to insure correctness. The most difficult part of the work is now 

 completed. An appropriation has been made by the Institution for defray- 

 ing the expense of the clerical labor which is required in preparing the 

 tables and performing the other laborious arithmetical calculations 

 necessary in reducing the abstract mathematical results to practical use. 



Among the papers accepted for publication are three by Major 

 General J. G. Barnard, United States Army. The first of these relates 

 to the "Precession of the Equinoxes and Nutation as identified with the 

 phenomena of the Gyroscope." All writers who explain the "Preces- 

 sion" in a manner intended to be more or less adapted to popular com- 

 prehension, assume or demonstrate certain elementary facts which are 

 common to the general phenomena of the "Precession," and the move- 

 ments of the philosophic toy, the "Gyroscope." The intention of this 

 paper is to identify the phenomena, and to show that a common analysis 

 leads, when properly adapted to the different circumstances, to their 

 solution. As a matter of course, the introduction of the proper expres- 

 sions for the external forces into the general equations of rotating bodies 

 will give the particular equations for the special cases. In the 

 Mecanique Celeste, are thus derived the expressions for precession and 

 nutation-; but the analytical process is difficult, and the point of identi- 

 fication of the phenomena with those of the gyroscope is, in this point 

 of view, too remote to be interesting,, In this paper, solutions primarily 

 obtained for the gyroscope are subsequently made use of to develop all 

 the facts of precession and nutation. By the methods employed it is in- 

 cidentally shown that the jmenomena of "deviation" in rifled projectiles 

 may be explained. 



The second paper by General Barnard is on the motions of a "freely 

 suspended pendulum," and differs from other well known discussions of 

 this problem iu giving, as a preliminary, a simple explanation of the 

 origin of the forces which, on the surface of the rotating earth, cause a 

 progressive azimuthal motion of the plane of vibration ; and furnishing 

 the analytical expressions for these identical with those of Poisson 

 obtained by other processes. These same expressions exhibit forces 

 which more or less sensibly affect all motions of material bodies on 

 or near the earth's surface, as e.^g. the tidal currents, the winds, 

 the trajectories of projectiles, &c. The expressions obtained for the 

 pendulum are developed with much greater detail than has been done 

 in previous works. The differential expressions for the vibrations 

 of the so-called "spherical pendulum" are integrated by development 

 into series, arid expressions approximately accurate, obtained for the 



