AND THE FIGURE OF THE EARTH. 81 



Restoring the value of a, we get 



cPs . ds ^ X-Y . s.n 1 

 rfXrfY + rfX^^"~2~ + ^ y__Y = 0, 



which is integrable ; and its integral is -^ -\- n . s tan ?-=^ = some arbitrary func- 

 tion of Y, as X Y. Integrate again, then 



where %J/ is arbitrary. Effecting these integrations, and reducing, 



, = cos-'"^(/cos^»^.xYrfY + ,^x). 



To return to P„, we have the following systems of equations : 

 P.=/i>rfY, 



V = B J 2 [fuJ 2 £/Y;=C08 '^L_^y^C0s2^^rfY, 



^= cos-^^/^cos^^.rfY, 



f= cos-^^-^^^/.cos^^-^^^.rfY, 



*= cos 



whence 



P„ = ....ycos-2^5^^cos-2X^(y2os2«^^:^YrfY+^^x)rfYrfY...(/ltimes^^ 



Now cos ^ = cos (y-zn log ^J\^^ = ^ (\/|37^::+ x/It^)' 



Y — X 1 



and cos^ —^ — = , _ a ; and the complete integral will be expressed, by substituting 



for X and Y in terms of /a and <p. 



6. But an important point yet remains to be determined. The original equation, 

 being a partial differential equation of the second order, can only involve in its in- 

 tegral two arbitrary functions. But here, after % Y and -^ X have come in by two inte- 

 grations, we have n integrations to perform with respect to Y. It would seem, there- 

 fore, that no constant or arbitrary function of X must be added in these integrations. 



MDCCCXLI. M 



