Prof. Gauss on a ?iew general Principle of Mechanics. 137 



Of these quantities to coincides exactly with the value of the 

 same angle found by M. Puissant (arc z\ p. 45) ; but owing 

 to an error of calculation in the Conn, des Terns, there is more 

 than a minute of difference between ml and the angle v" at 

 p. 46. 



The inspection of the preceding formulas is sufficient to 

 prove that whatever be the situation of the stations, sin (to — 

 ju/) may be reckoned equal to sin (to' — [x!) ; and consequently, 

 to + to' on the spheroid to i^ + fjJ on the sphere : the difference 

 arising from the variation of latitude is of the order e 4 , and 

 is always insensible. 



Jf we put M and M' for the angles which the geodetic curve 

 makes with the meridians of the two stations, the true azi- 

 muths to and ml will be indistinguishable from M and M' so 

 long as the two stations are so near one another that the curve 

 between them may be reckoned in one vertical plane. As the 

 distance between the stations increases, M and M' separate 

 themselves from m and to', and M-fM' is no longer equal to 

 to -f to' nor to ju, + f//. It may be added that to and to' are exactly 

 equal to p and jw.' when the two stations are equally distant from 

 the equator : but in the like circumstances, M and M' are not 

 strictly speaking equal to ju, and j«/, the difference, although 

 extremely small in most cases, increasing as the distance of 

 the stations increases. Precision of ideas seems to require 

 that in such investigations a distinction should be made be- 

 tween the true azimuths m and to', and the angle M and M'. 



July 13, 1830. . J. Ivory. 



XVI. On a new general Principle of Mechanics. By Pro- 

 fessor Gauss of Gottingeri* . 



TT is well known that the principle of virtual velocities con- 

 A verts the whole science of statics into a mathematical 

 problem ; and by D'Alembert's principle the theory of dy- 

 namics is reduced to that of statics. It is therefore evident 

 that there can be no new fundamental principle of the theory 

 of motion and equilibrium but what is contained in those two 

 others, and may be deduced from them. This is, however, no 

 reason why every other new principle should be of no value. 

 It will always be interesting and useful to take new views of 

 the laws of nature; some problems may thereby be more easily 

 solved, or a particular fitness may be discovered in them. 

 The great geometrician, who has raised the structure of me- 

 chanical science on the principle of virtual velocities in such 



* Translated from the original German, in Crell's Journal of Mathematics. 



a brilliant 



