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LXV. On the Figure of the Earth, as deduced from Measure- 

 ments of the Meridian. By J. Ivory, Esq. M.A. F.B.S.* 



TN my last communication I have shown that the five por- 

 •*■ tions of the meridian which have been most exactly mea- 

 sured, belong to one and the same elliptical spheroid, although 

 they occupy very dissimilar situations on the earth's surface. 

 The almost perfect precision with which the elliptical elements 

 agree with the actual measurements, is indeed not a little re- 

 markable, and seems hitherto to have escaped notice. In re- 

 turning to this subject, which is of great importance in the 

 question about the figure of the earth, my intention is to dis- 

 cuss it more thoroughly, by examining it in every point of 

 view that may throw light upon it. 



The equations deduced from the five measurements are as 

 follows (Phil; Mag. for May, pp. 345, 346): 



188510= A (3*117500— 6*2261. s + 3*095. s 2 ) 

 598630= A (9*895891 — 18*1985. e + 6*163. e«) 

 751567 = A (12*370302— 6*2811 . £-10*466. e 2 )f 

 172751 =A (2*840172— 0*3844. e— 2*166. e 2 ) 

 98870 =A (1*622022+ 0*8378. e- 0*022. e 2 ). 



In solving these equations I before supposed that the ele- 

 ments sought were approximately know r n; but I shall here 

 proceed by a more direct method, which is independent of any 

 previous knowledge of the elements. For this purpose, as- 

 sume, A=R(1 + As + Bc 2 ), 

 R, A and B being quantities to be determined : substitute the 

 value of A in every one of the five equations, neglecting the 

 powers of e above the square ; and, having added all the re- 

 sulting expressions into one sum, equate the coefficients of e 

 and s 2 to zero : then, 



1810328 = 29*845887 X R ; R = 60655*9 



29*8459 X A = 30*2523, A = 1*0136 



29*846 B = 30*252 A + 0*396, B = 1*141 

 and, consequently, 



A = R (1 + 1*0136 . s + 1*141 . g*). 



It is obvious that R is the radius of the circle in which the 

 sum of the lengths of all the arcs answers to the sum of their 

 amplitudes. The element A now depends upon e, which is 

 the only unknown quantity. In order to find e, substitute the 

 value of A in every one of the five equations, bringing all the 



* Communicated by the Author. 



f In the Phil. Mag. for May, the coefficient off* is 6-266; but the inad- 

 vertence does not affect any of the results. 



terms 



