1865.] On Local Attraction. 37 



e= — ^9 { 1+0-0608^+0-0085^ +0-0103^3+0-0016^+0-005^ 5 



-f 0-0008^ -f 0-0039*, -f 0-001639i 8 }. 

 where tj t- ... t 8 are the eight unknown deviations of the plumh-line 

 from the true vertical at the standard stations of the eight arcs arising 

 from local attraction. 



These formulae for the semiaxes and ellipticity of the mean figure 

 of the earth show us, that the effect of local attraction upon the final 

 numerical results may he very considerable : for example, a deflection 

 of the plumb-line of only 5" at the standard station (St. Agnes) of the 

 Anglo- Gallic arc would introduce a correction of about one mile to the 

 length of the semi-major-axis, and more than three miles to the semi- 

 minor-axis. If the deflection at the standard station (Damargida) of 

 the Indian Great Arc be what the mountains and ocean make it 

 (without allowing any compensating effect from variations in density 

 in the crust below, which no doubt exist, but which are altogether 

 unknown) viz. about 17" - 24, the semiaxes will be subject to a 

 correction, arising from this cause alone, of half a mile and two miles. 

 This is sufficient to show how great a degree of uncertainty local 

 attraction, if not allowed for, introduces into the determination of the 

 mean figure. As long as .we have no means of ascertaining the 

 amount of local attraction at the several standard-stations of the arcs 

 employed in the calculation, this uncertainty regarding the mean 

 figure, as determined by geodesy, must remain. The effect of our 

 ignorance in this case is far more serious than that already noticed in 

 mapping a country with minute precision. 



4. The third section of the Paper is occupied in devising means 

 for removing this ambiguity. Although it has been necessary to 

 assume one step in the argument, I think that the sequel shows that a 

 very high degree of probability exists that the process is a correct one. 



Each of the three great arcs — the Anglo- Gallic, the Russian, and 

 the Indian — is divided into a number of subordinate arcs. I therefore 

 take each of these three great arcs and apply the method described in 

 the last section to find the semiaxes of the ellipse which best 

 represents that arc. The expressions for the semiaxes involve one 

 unknown quantity, viz. the amount of deflection at the standard 

 station of the arc. In this way I obtain the semiaxes of three ellipses, 



