AND THE FIGURE OP THE EARTH. 79 



From (4.), by subtraction, 



whence 



_ ,-(x-Y)>/:rT__^ 



and 



1 — |U*^ = 



2 



consequently 



^+ "^"lx-Y = <'- Let»(n + 1) = «. 



Let 



4 cos^ 



<fP„ • , <^t> , aP„ ^ jn rf«^ 4 „X-Y 



^« = t;,then^+ ^X-Y = ^^^^^^n = " ^x IT^Q^'-T- 



4 cos' 



2 



JP„ rZ^^, 4 X-Y dv 4 X-Y . X-Y 



or 



Let 



Then 



and 

 whence 



d^v ,dv X-Y av 



dXdY"^ dX^^^ 2 +. aX-Y — "- 



4 cos* 



rfo , ^ X-Y ^ 

 ^ + t;tan— 2- = ^. 



<Z'v dv X-Y , t? rf^ 



<ZXrfY + fZX^^^ 2 +„^oX-Y-rfX' 



2 cos* — - — 



dt , a~2 V ^ 4 „X-Yrf/ . 



^ + -4 rx-Y = ^' or«;=~j3r2COs2-2-Jx' 



cos 



2 



rftj 4 ,X-Y rf3^ 4 X-Y . X-Y<f/ 



dY=T=r2''''^ 2 rfXrfY-J^I^^^^s-^-^'^'-^-d^' 



4 ,X-Y </*^ 8 X-Y . X-Y dt 



^ = - ^ITi C^S2 -^- jX^ - ^-^ COS -2- Sin -2- rfX' 



or 



^^ . ^^of X-Y . , g-2 _^ 

 ^X5Y + ra ^ ^^^ T" + 4 003* ^"^ "" 



2 



Let 



^^ . o.. X-Y 



