:h59 



§ 2. The introduction of polar coördinales, clioosiiig as origin (lie 



centre of curvature at the top M (tig. J), gives the advantage that 



rr 

 there is no discontinuity at I'J = — ; in that case 



.V ■=. Q .-iin «I and // = /^o — {) cos {) . . . . . (4) 

 and the equation (1) becomes 



p sin i)- — o' cos i> o' f 2o"— op" 2 



— ^ -4- " " ~ = } /c(lL — (,co,iO). . (5) 



If we now put 



i)=R^(\ — t) and Tr=T,+T, + T, f ..... . . . (4') 



where t,, t,, etc. represent the successive approxiuiations to the 

 infinitely small quantity t, we can, as long as r and t' are infinitely 

 small, separate equation (5) into a series of other ones, the first of 

 which being 



t\ sin i)-\-r\ cos ^4-2r, sin {h=kR^' (I — cos i)) sin i) ', . . (5') 

 hence *) 



'1 I cos » 



r 1 leR ' 



{I— COS i)) \ 2 cos ih log 



n {cos ^Y 



(6) 



an expression which remains valid from /> =0 to V = ;rr throughout. 



§ 3. The result of the third approximation is as follows 



«=|jrR/(l — cos ay (1 4-2 cos 0) f i^rkR^' (I — cos i)y cos'th^ 



, /I f cos ih\ 

 4 ^JikR,' sin' .*> (1 — 2 CUV {)--\-2 cos* i)) log i — ~ ) . . (7) 



and 



V ^^inR,* (1 — cos iyy (2 + cos ,*^) — I jiIcR^' (\ — cos i))" (2 + cos i»)) — 



/I + cosi)-\ 

 ^nkR,' sin' i) log i -^ J (7') 



^ 4. Between the angles '9 and fp the following relation holds: 

 Q sin (') — {i ros I'J 



S171 'f = 



|/p'+i>" 



=:: flu & f t' cos «y \- 



^) In order to integrate these equations we have to bear in mind, thai 

 cos Ö- (t" sin d- -}~ r' cos & -f- 2,x sin ti ) = — \sin «^ (t sin d \ t' cos &) \ 



and 



, d f X \ 



X smd" -\- X cos >> =r co.v'' tr — j, 



dH \cos >') I 



The integration doe.s not ofTer any special difïicullieSi but the calculations are 

 long, that of Tj being already very laborious; for that reason we have confined 

 ourselves to Tj. 



It is easily seen, that RqTx = Zi cos B. 



