Prof. Gauss on a new general Principle of Mechanics. 139 



square of the deviation of every point from its free motion into 

 its mass. Let m, m', m!', &c. be the masses of the points ; a, 

 a\ a", &c. their places at the time t ; b, 6', b", &c. the places 

 which they would occupy if entirely free in their motion after 

 the infinitely small particle of time dt, in consequence of the 

 forces acting upon them during- this time, and of the velocities 

 and directions acquired by them at the time t. Their real 

 places c, c', c", &c. will then be those for which of all places 

 compatible with the conditions of the system the quantity 

 m(bcY + m' (&' c') 2 + m" (b" c") 1 &c. is a minimum. The equi- 

 librium is evidently a particular case only of the general law, 

 and the condition for this case is, that m {ahf +m' (a' b')' 1 + 

 m" (a" b")- &c. itself is a minimum, or that the continuance of 

 the system in a state of rest more accords with the free motion 

 of the single points than any possible change which the system 

 could undergo. Our principle is easily deduced from the two 

 others in the following manner. 



The force acting on the material point m is evidently com- 

 posed, first, of one which in conjunction with the velocity and 

 direction existing at the time t, will carry it during the time dt 

 from a to c; and of a second one, which would carry it in the 

 same time from a state of rest in c through c b, considering 

 the point as free. The same applies to all other points. 

 According to D'Alembert's principle, therefore, the points 

 m, ml, m", must, by the conditions of the system, be in equi- 

 librio in the points c, c', c", &c. when acted upon by no other 

 forces but the second ones, tending to c b, c' b', c" b", &c. Ac- 

 cording to the principle of virtual velocities the equilibrium 

 requires that the sum of the products of every three factors, 

 viz. of each of the masses m, m', m", &c. of the lines c b, c' b', 

 c" 6", &c. and any motions, compatible with the conditions 

 of the system, projected on those latter lines respectively, 

 should always be == 0, as this principle is commonly enun- 

 ciated*, or more correctly, that that sum should never be posi- 

 tive. If, therefore, y, y', y", &c. are places, compatible with the 

 conditions of the system, different fromc, c', c"; and S, .&', $% &c, 



* The usual enunciation of the principle tacitly supposes such condi- 

 tions, that the motion contrary to every possible motion should likewise be 

 possible, as for example, that a point must necessarily remain on a certain 

 surface, that the distance of two points from one another should be always 

 the same, &c. But this is an unnecessary restriction not always accordant 

 with nature. The surface of an impenetrable body does not force a ma- 

 terial point on it, always to remain on it, but only prevents its deviating 

 from it on one side; a stretched inelastic but flexible thread between two 

 points renders impossible an increase of distance, but by no means a di- 

 minution of it, and so on. Why not therefore at once express the law 

 of virtual velocities so as to embrace all cases ? 



denote 



