134 CELESTIAL MECHANICS. 1.11. 11. 



280. Theorem. The oscillations of a 

 gravitating body, moving freely on a spherical 

 surface, of which the radius is r-, are performed 



m a time l-.^-. .^-^p-'J l + yv+ (g;3> 



13 5 



r* + (-^Jq) V^ + • . • ; t being the semicircum- 



ference to the radius 1, a the greatest and b 

 the least distance of the body below the 

 centre, g the space described by a heavy 



body in the unit of time, and y^= t-tt^i — -p* 



In this case we obtain from the equation lS^s-\-R^rzzOj 



compared with P = ^, Q- -^, and jR = -^ (264), 



.1 .1 .. -rx <^dj7 ^ X ^ My y , 



the three equations = -r— + A -, Ozz -j-f + A — , and 



ddz z 

 0=:-T-j + A gy A being the pressure on the surface ; 



CT X cT y 

 for since r^nar^ + v^ + z^ we have -^r-^ — ,-^ =: —.and 



hx r hy r 



gV 2 ^^ . „ djr2 + d?/2 + dz2 ^ ^, ^^ 

 s-=:-. Now since v^zz -^^ :=/ (Pd^ + Qdy + 



JRdjz), P, Q, and R being the accelerating forces concerned 

 (264 Cor.), the fluent here becomes v^zz2fgdiZ—c + 2gz, 

 consequently the pressure derived from the centrifugal 



force will be simply ^ (272 Schol.) to which adding 



8V z 

 the force of gravity, reduced in the ratio -^j or — , that is 



gz . c -\- Sgz 



— , we obtain —: — ^— •, for the whole pressure on the 



surface. 



