4 REPORT — 1842. 



ence of/', taken with respect to this variable. Let/" = const, be a second integral. 

 By means of this equation I eliminate a second variable, and I call p" the partial dif- 

 ference of /" with respect to this variable. Let us suppose that we know all the in- 

 tegrals of the problem but one, and that with respect to each integral /= const, we 

 seek the corresponding partial difference^ with respect to the variable, which we elimi- 

 nate by means of this integral. The number of variables exceeds by unity that of 

 the integrals : we eliminate by means of each integral a new variable, and we thus 

 succeed in expressing all the variables by means of two of them. Let us call these 

 two variables x and y and x' and y', their first differentials taken with respect to the 

 time. We shall express by means of .r and y the quantities x' and y', as well as all 

 the quantities p', p", &c. : since x' and y' are the first differentials of x and y taken 

 with respect to the time, we shall have the equation 

 y' dx — x 1 dy — 0, 

 where x' and y' are known functions of the two variables a- and y. It is this differen- 

 tial equation, the last of all of them, which we must integrate in order to obtain the 

 complete solution of the problem. Now I show that on dividing this equation by the 

 product of the variables p', p", &c, its first member becomes an exact differential, and 

 therefore the integration of this equation is generally reduced to quadratures. 



When we have any system whatsoever of material points, the simplicity of the pre- 

 ceding theorem is in no respect altered, provided we give to the dynamical differen- 

 tial equations that remarkable 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 general researches of analytical mechanics. It is 

 true that the formulas of Sir W. Hamilton are referrible only to the cases where the 

 components of the forces are the partial differences of the same function of the coordi- 

 nates; but it has not been found to be difficult to make the changes which are neces- 

 sary in order that these formulas may become applicable to the general case, where 

 the forces are any functions whatever of the coordinates. 



When the time enters explicitly into the analytical expressions for the forces, and 

 into the equations of condition of the system, the principle of the final multiplier, 

 found by a general rule, is applicable also to this class of dynamical problems. There 

 are also some particular problems into which enters the resistance of a medium, which 

 give rise to similar theorems. It is the case of a planet revolving round the sun in a 

 medium whose resistance is proportional to any power of the velocity of the planet. 



The analysis which has conducted me to the new general principle of analytical 

 mechanics, which I have the honour to communicate to the Association, may be ap- 

 plied to a great number of questions in the integral calculus. I have collected these 

 different applications in a very extensive memoir, which I hope to publish upon my 

 return to Kbnigsberg, and which I shall have the honour of presenting to the Asso- 

 ciation as soon as it shall be printed. 



Extract from a Memoir entitled " Considerations on the Principles of Ana- 

 lytical 3Iechanics." By Professor Braschmann of Moscow. 



The principle of virtual velocities, on which is based the theory of equilibrium 

 and of motion, has not, in my opinion, been explained in a manner which is clear 

 and unobjectionable ; and I am also inclined to believe that the problem of equili- 

 brium has not been treated analytically in a point of view sufficiently general, and 

 that there are still many observations to be made on the correctness of the applica- 

 tion of the principle of virtual velocities to certain problems. 



Similar observations may be made also with regard to the theory of motion. M. 

 Ostrogradsky brought forward, some years ago, some new and general ideas on the 

 laws of equilibrium and of motion in two memoirs, one of which bears the title, " On 

 the Momenta of Forces ;" and the other, " On the instantaneous Displacements of 

 the points of a System." Profiting by his enlightened views, I published, in 1837, 

 a treatise in the Russian language on the equilibrium of solid and fluid bodies, from 

 which I will now give a very short extract relating to the method I have there fol- 

 lowed, and I shall add some observations which escaped me at the time of the pub- 

 lication of that work, respecting the number and the character of conditions of 

 equilibrium. 



