TRANSACTIONS OP THE SECTIONS. 3 



whenever the forces are functions of the coordinates of the moving points only, and 

 the problem is reduced to the integration of a differential equation of the first order 

 of two variables, it may also be reduced to quadratures." Now I have 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 generality very superior to that of other known principles, inasmuch as it 

 applies to cases in which, when the analytical expressions of the forces, as well as the 

 equations by which we express the nature of the system, are composed of the coordi- 

 nates of the moveables in any manner whatever, principles (such as the principle of 

 the conservation of living forces, of the conservation of areas, and of the conservation, 

 of the centre of gravity) are superior to the new principle in several respects. In, the 

 first place, these principles afford a finite equation between the coordinates of the 

 moveables and the components of their velocities, whilst the integral found by the 

 new principle is simply reduced to quadratures. In the second place, we suppose in 

 the application of the new principle that we have already succeeded in discovering 

 all the integrals but one, a supposition which will be realized in a small number of 

 problems only. This will be sufficient to convince us of the importance of the new 

 principle ; but this may be made still more manifest, if I am permitted to illustrate 

 its application by a few examples. 



1st. Let us consider the orbit described by a planet in its motion round the sun. 

 The differential equations in dynamical problems being of the second order, we may 

 present them under the form of differential equations of the first order, by introducing 

 the first differentials as new variables. In this manner the determination of the orbit 

 of the planet will depend upon the integration of three differential equations of the 

 first order between four variables. We find two integrals by the principles of living 

 forces (forces vives) and areas. The question is thus reduced to the integration of a 

 single differential equation of two variables and of the first order. Now, by my gene- 

 ral theorem, this integration may always be reduced to quadratures. If therefore we 

 choose to reckon this theorem amongst the other principles of mechanics, we see that 

 the general principles of mechanics alone are sufficient to reduce the determination of 

 the orbit of a planet to quadratures. 



2nd. Let us consider the motion of a point attracted to two centres of force, after 

 Newton's law of gravitation. The initial velocity being directed in the plane passing 

 through the body and the two centres of attraction, we still have to integrate three 

 differential equations of the first order amongst four variables, one integral of these 

 equations being furnished by the principle of living forces. Euler has discovered an- 

 other, and thus has succeeded in reducing the problem to a differential equation of 

 the first order between two variables. But this equation was so complicated, that any 

 person but this intrepid geometer would have shrunk from the idea of attempting its 

 integration and reducing it to quadratures. Now, by my general principle, this re- 

 duction would have been effected by a general rule without any tentative process, 

 without any extraordinary effort of the mind. 



3rd. Let us consider also the famous problem of the rotatory movement of a solid 

 body round a fixed point, the body being under the influence of no accelerating force. 

 In this problem we shall have to integrate five differential equations of the first order 

 amongst six variables. The principle of living forces gives one integral, that of areas 

 gives three others, and the fifth is found by my new principle. We thus see all the 

 integrals of this difficult problem found by the general principles of mechanics alone, 

 without our being required to write a single formula, or even to make a choice of 

 variables. 



I will endeavour now to enunciate the rule itself, by means of which the last inte- 

 gration to be effected in the problems of mechanics is found to be reduced to quadra- 

 tures, the forces being always functions of the coordinates alone. Let us suppose, in 

 the first instance, any system whatsoever of material points entirely free. Let there 

 be found a first integral /' = const., the variables which enter into the function /' 

 being the coordinates of the moveables, and their first differentials taken with respect 

 to the time. I avail myself of the equation 



/' es const, 

 for the purpose of eliminating any one of the variables, and I call p' the partial differ- 



b 2 



