170 Proceedings of the British Association. 



Mr. Hodgkinson also presented a communication on the mode 

 of conducting experiments on the resistance of air. 



On a new general principle of Analytic Mechanics } by Prof. 

 Jacobi of Konigsberg. In the different problems relative to the 

 motion of a system of material points which have been hitherto 

 considered, one may make, said the Professor, an important and 

 curious remark, " that whenever the forces are functions of the 

 coordinates of the moving points only, and the problem is redu- 

 ced to the integration of a differential equation of the first order 

 of two variables, it may also be reduced to quadratures." The 

 author has succeeded in proving the general truth of this remark, 

 which appears to constitute a new principle of mechanics. This 

 principle, as well as the other general principles of mechanics, 

 makes known an integral, but with this difference, that whilst 

 the latter give the first integrals of the dynamical differential 

 equations, the new principle gives the last. It possesses a gener- 

 ality very superior to that of other known principles, inasmuch as 

 the analytical expressions of the forces, as well as the equations 

 by which we express the nature of the system, are composed of 

 the coordinates of the movables in any manner whatever. The 

 Professor proceeded to show the application of his principle, and 

 its advantages in the following problems, viz. the orbit described 

 by a planet in its motion round the sun ; the motion of a point 

 attracted to two centres of force, after Newton's law of gravita- 

 tion ; and the problem of the rotatory movement of solid bodies 

 round a fixed point. He then enunciated the rule itself, by 

 which the last integration to be effected in the problems of me- 

 chanics, is found to be reduced to quadratures, the forces be- 

 ing always functions of the coordinates alone ; and observed,, 

 that when we have any system whatsoever of material points, 

 the simplicity of the preceding theorem is in no respect altered, 

 provided we give to the dynamical differential equations, that re- 

 markable form under which they have been presented for the 

 first time by the illustrious Astronomer Royal of Dublin, and 

 under which they ought to be presented hereafter in all the gen- 

 eral researches of analytical mechanics. It is true that the for- 

 mulas of Sir W. Hamilton are referable only to the cases where 

 the components of the forces are the partial differences of the 

 same function of the coordinates ; but it has not been found to 

 be difficult to make the changes which are necessary, in order 



that these formulas may become applicable to the general case 



