14 Archdeacon Pratt on the Figure of the Earth, 



tion to be added to the observed latitude of A to obtain the point 

 a in the variable ellipse, that is, the arc K a, 



has, then, to be found. 



If the earth had its mean form, the plumbline at A would hang 

 in the normal I A to the mean ellipse. Hence the angle which 

 the plumbline at A makes with /A is the change in the latitude 

 of A arising from local attraction, and therefore equals t. This, 

 then, is the angle between the normal at K and the line I A. 

 Add to this the angle I AX, and we have the whole angle mea- 

 sured by the arc K X. 



This angle I AX can be found by conic sections as follows : — 

 Draw A E, ZF, XG perpendicular to OH, the semiaxis major, 

 and produce the normals / A, \A to P and Q. Let e and e' be 

 the ellipticities of the mean and variable ellipses. Then by conic 

 sections OP = 2e . OF, OQ = 2e' . OG. Hence, neglecting small 

 quantities of the second order, 



7aa i. /*rkxj it>ij\ cot APH— cot AQH 



tan IAX= tan (AQH — APH) = - . „„ txtttt 



v ' 1 + cot APH . cot AQH 



_ (PF-QF)AF _ (OQ-OP)AF 

 ~AF 2 + PF.QF~~ AF 2 + OF 2 



= 2(e' — e) sin / cos 1= (e' — e) sin 21, 



I being the observed latitude of A ; 



1" 

 .-. IAX= (e'-e) sin 21 -.— -^ 

 v ' sin 1" 



or, by formulae of paragraph 3, 



.-. K\=* + n(v — V). 



Hence, finally, 



%=t-\-n(v — Y)+z 



v ' i i 

 and 



dec dx _ 



du~~ ' dv ~ 



7. We are now ready to apply the principle of least squares to 

 discover the form of the mean ellipse. By paragraph 3 the sum 



