256 Analysis of the Mecauique Cdesfc of M. La Vlace. 



(ksic line is nerpendicvilar to the corresponding plane of the 

 terrestrial meridian, and he obtains an equation whieh de- 

 termines the diflerence of latitude of the two extremities of 

 the are. It is very remarkable that the function which 

 gives this difference is equal to t!ie azimuthal angle ol)served 

 at the extremity of the same arc, measured in the direction 

 of the meridian and divided by the tangent of the latitude 

 at the first point of this arc. This function may therefore 

 be determined two ways; and we might judge if the values 

 found, whether of the dilTerence of the latitudes or of the 

 azin)ulhal angle, are owing to the errors of observations or 

 to the eccentricity of the "terrestrial parallels. The author 

 afterwards calculates the difference in longitude of the two 

 extremities of the arc measured in the direction of the pa- 

 rallels, as well as the azimuthal angle formed by the ex- 

 tremity of this arc with the corre'^ponding plane of the ce- 

 lestial meridian. Finally, he determines the osculating 

 radii of the geodesic lines, whether drawn in the direction 

 of the meridian or in the direction of the parallels, and he 

 deduces from it (hat of the geodesic line, which forms with 

 the meridian any given angle. Considering afterwards 

 the osculating ellipsoid, the author teaches us how to de- 

 termine it from the measurements of the earth. 



We have previously seen that the elliptic figure must 

 be that of the earth and the planets, supposing them to 

 have been originally fluids, if in other reipeCts, by becom- 

 ing hard, they have preserved their primitive figure: it was 

 therefore natural to compare with this figure the measured 

 degrees of the meridian ; but this comparison has given for 

 the figure of the terrestrial meridians difierenl ellipses, and 

 which are too far removed from the observations to be 

 admitted ; whence it follows that the figure of the earth is 

 more complex than had been at first supposed. Neverthe- 

 less, before abandoning the elliptic figure entirely, it is im- 

 portant to determine that in which the error \i smaller than 

 in any other of the same nature. The author gives two 

 different methods for attaining thi< oliject; the first is gene- 

 rally applicable at all times, when we have a certain numbef 

 of observations, which we suppose to be represented by a 

 function wh^se form is given; it is requisite to determine 

 this function m such a manner that the errors of observa- 

 tion may be less than in any other of the same form. 

 Uavino, for instance, any given number of obscrvatiofis of 

 a comet, we may bv their means determine the parabolic 

 orbit in which the greatest error is (abstracting from the 

 sign of the error) le?s than in any other of the same mature: 

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