338 ON THE MOTION OF 



this problem which have since been given, there is perhaps 

 none equal to Euler's in directness and perspicuity. The 

 methods of obtaining the resulting formulas have however 

 been, with great advantage, occasionally modified so as to suit 

 particular views and purposes. It is in astronomy more es- 

 pecially that these three elements of aspect are most employed, 

 for which reason they are preferred by Laplace to the three 

 indefinite integrals, the angles P, Q, i?, notwithstanding the 

 greater symmetry which arises from the use of these three 

 angles. It is to Euler also that we are indebted for formulas 

 which lead to this last determination, by which the cosines 

 of the nine angles are made to depend by the medium of dif- 

 ferential equations on the values of the integrals P, Q, R. In 

 the sixth volume of the Berlin Transactions for the year 1750, 

 in a memoir entitled Decouverte cVun nouveau principc de me- 

 canique, Euler gave the formulas, now so well known, which 

 express the motion of every point of a system in terms of the 

 coordinates of the point and the motion of progression and 

 rotation common to all the points. These expressions were 

 employed by Lagrange in obtaining the relations by which 

 the variations of the cosines of the axe-angles were reduced to 

 the three variations dP, dQ, dB, or the three analogous varia- 

 tions of the angles of rotation round the axes fixed in space. 

 Finally, in the Memoirs of the Academy of Turin for the 

 years 1784 and 1785, there is a curious paper by Monge, in 

 which, having occasion to introduce these nine cosines, he 

 takes for the independent variables the three angles x 0,x„ 

 x'0,y„ x"0,z„ and gives without demonstration the values of 

 the other six, expressed in terms of these three. Lacroix has 

 inserted these results, with an accompanying demonstration, 

 in his quarto treatise on the Differential and Integral Calculus ; 

 but I am not aware that this method of determination has 

 been employed in Analytical Mechanics. 



One of the methods by which the transformation from indi- 

 vidual to common variations has been effected is founded on 

 the formulae which give the variations of the cosines a, a', a", 

 &c. in terms of the variations of the angles of rotation round 

 the space-axes. This method has the advantage of leading 



