136 CELESTIAL MECHANICS. I. ii. 11. 



r^^a^r^^abr^—b^r^ ah r^ + a ^b^\ 

 ^^( aT^ + aTV—) = ^ ^• 



r^—a^r^—b ^r^+a^h2 ,^ . . u ^ 4i, wi. 



-^ . It must be observed that tne 



a-\-b 



quantities a and b will be the greatest and least values of 



z ; [for otherwise the fluxion rdz would not vanish, as it 



must do, when the curve becomes horizontal], 



a — z 

 Making now sin 6z= ^ 7, we have d sin 5z=cos 6d9zz 



— dz - . z — b . 



and smce cos zz ^f r» d d zz 



2s/{a—z) V (a— 6) ' ^ a—h* 



— dz a — b _ — dz 



2V(a— 2) s/ {a—h)' ^ z—b "" 2V_(a— z) s/\z—h)'' 



consequently, in the ascent of the body, 



— rdz , 2rd5 



- At — ——^ —.Now 



s/(a— z)^/(z— &)V(2yz+/) - >s/{2gz-^f) 



a — z 

 since sin* d= ^, (a — b)^m^z=.a — z,zzza — {a — &) sin^ 5, 



j'2 -\-ab 

 andy being = 2g. 7-, 2gz+fzz2g(a — (a — 6) sin 25 + 



r«+a6v ^ a^A-ab + r^+ab—Ca^—b^) sin 29 



—T— ) = 2 cf --: , and 



a+b J ^ a+b 



a^—h2 r 



makmff — — ft^^ y^> we have at zz >J — . >J 



^ a^-\-r^-j-2ab 9 



2r{a-\-b) dd 



a2+r2+2a6 V(l— y^sinsfi) 



CI- • ^ ^ — ^ 1 z — b 



bmce sin^ Q zz -• and cos ^e~ 7,we havea cos^ 



a — a — b 



^ J . „ az — ab + ab — bz , 2; .„ , , 



+ 6 sm20= =zz, and — will be the cosme 



a — 6 r 



of the inclination of the radius to the vertical diameter. 



If -23- be the angle made by the revolving vertical plane of r 



and z,with the plane of a^andz, we have ta 'mzz ^ and dta-io-zi 



