K D. Preston — Measurement of th\ Peruvian Arc L6 



change for the supposed error, but it must be remembered that 

 Dot only is the arc at the equator and therefore has great influ- 

 ence in tlic determination of the elliptic figure, but also that it 

 comparatively short arc and hence any error in the ampli- 

 tude has a proportionately greater effect on the length of a 

 degree deduced therefrom. 



The individual influence of arcs where many enter into the 

 determination should not, however, he overestimated. If we 

 suppose ares of one degree to be measured from the pole to 

 the equator, say 10° apart, their weights in fixing the polar 

 axis are approximately as the numbers 39, 43, 54, TO, 89, 111, 

 131, 1-16, 157, 161 and in the determination of the equatorial 

 axis these same numbers apply in an inverse order. A curve 

 plotted on rectangular coordinates, with the earth's radii and 

 the above weights as arguments, has a point of inflection in 

 middle latitudes, and since the ellipticity is unity minus the 

 ratio of the two axes, middle arcs have very little influence on 

 the ellipticity. 



The pendulum observations indicate that the density of the 

 mountains is about one-fifth the mean density of the earth. 

 We may therefore assume that the Andes in the neighborhood 

 of Quito are one-half as dense as the general surface of the 

 earth ; and if we take 15" for the deflection at each end of the 

 arc the ellipticity of the figure is changed by about one-fourth 

 part of itself. 



The effect of any change in an equatorial arc, on the figure 

 of the earth, as deduced from the nine arcs above mentioned is 

 easily found. The conditional equations are combined by least 

 squares in order to find the values of M and N in the equation, 



tf=M + 1ST sin 2 / 



where d is the length of one degree in latitude Z, M is the 

 length of one degree at the equator, and N is the difference in 

 length between the equatorial and polar degree. The change 

 in length of the equatorial degree w r ill be given by differentiat- 

 ing an expression of the form 



2 y (2 a - 2 ab) 

 9 2 b 3 -(2 b) 3 



where a is the independent variable. The degree being at the 

 equator, the differential of 2 ab is zero, and the change in the 

 length of the equatorial degree from the solution of the normal 

 equations would be about two-thirds the assumed linear error 

 in the individual arc. Knowing the differentials of M and N 

 the changes produced in the eccentricity and ellipticity are ob- 

 tained without difficulty. 



