410 Dr. Schmidt's Deiermination of the Dimensions of the Earth, 



application of the formulae derived from this assumption, to the 

 different measurements which have been executed, has proved 

 that every combination of any two of them gives a different 

 result for those dimensions. It became, therefore, necessary 

 to apply to the whole mass of the best measurements, a prin- 

 ciple accordant with the nature of the subject, by which the 

 most probable figure of the earth's surface, resulting from 

 all of them, might be elicited. This has partly been done 

 in an unfinished paper by Walbeck, entitled " Disscrtatio 

 de forma et magnittidinc telluris, ex dimensis arcnhis vieri- 

 diani definiendis." Walbeck has founded his researches on 

 the principle that the sum of the squares of the differences be- 

 tween the measured and calculated arcs should be a minimum. 

 But he neglected in these calculations the second power of the 

 ellipticity, and likewise the partial arcs determined between 

 the extreme points of a measurement; and his results were 

 therefore susceptible of some improvements by a more accu- 

 rate investigation. 



Having been induced by Professor Gauss to take up this 

 subject, I have taken into account the second power of the 

 ellipticity, and have so far changed the above-mentioned prin- 

 ciple, as to make the sum of the squares of the differences 

 of the calculated and observed latitudes a minimum, instead 

 of that of the squares of the differences of the measured and 

 computed arcs of the meridian. The results of my calcula- 

 tion have been published by Prof Gauss in his book entitled 

 " Determination of the Difference of Latitude of the Observa- 

 tories of G()ttingen and Altona." But on a revision, a small 

 error in calculation was discovered; and I became acquainted 

 with a small correction which is to be applied to the English 

 and East Indian measurements (see Phil. Trans., where Capt. 

 Kater has proved that the lengths of the arcs in the East Indies 

 are to be multiplied by 1 — 0*00001 8, and those in England by 

 1 +0*000007). A new calculation therefore became necessary, 

 which I here communicate. 



Lety denote the length of the mean degree (the three-hun- 

 dred-and-sixtieth part of the terrestrial meridian), a its ellipti- 

 city; so that iffi represent the semiaxis major, and b the semi- 

 axis minor, we have b = [\ —a.) a. Let it be assumed, that 



f = toises, and « = ^ ; then the following 



"^ 1000 



measurements give these final equations for determining u 

 and 3/: 72*13 = 2100-90 ?< + 1 763-1 Gj/ 



200-77 = 1763*16 ?f + 9318-66 J/ 



Hence 



