304 On a Correction to Observed Latitudes. 



Now let X, Z be the components of the iittraction of the 

 earth at P parallel respectively to the equator and to the polar 

 axis, F the centrifugal force at P ; then, (f> being the latitude 

 of P, we have the necessary relation (X — F) tan c/>=Z. At 

 F, instead of X-F and Z, we have X-F + ^(X-F) and 

 Z + 3Z ; and these will correspond, not to tan but to 

 tan {cj) + S<p). Hence, differentiating, we have 



(SX — SF) sin <^ — 8Z cos ^ = — Z cosec (f>B<f>. 



dY dY 

 Now X= — -J-,, and Z = ^ ; and if we put n for the height 



P P^, wc have in passing from P to P^, 



Sf= n cos ^, hi I — n sin 0, 

 and 



— 6 A = /I cos <p -r^K- + « sni -77-,T? 



— b/j = n cos ffi .. „ + n sni — y^ ? 

 whence 



^^^^8c/, = smc/>cosc^ (^^ - ^j + ^^(sm^0- cos^^ 



+ ft)^ sin </) cos (^. 

 The values of the differential coefficients are found to give 



where »'^ =/^ + A^. Also 



cos^={(l-2eJ). 



Substituting these in the expression for 8^, the result after 

 some little reduction is 



Z ,. M 



= n^^/A(Ge-8(.~^) + ;.) 



sin 

 Xow, with sufficient approximation, 



