336 Prof. Cayley on the Geodesic Lines 



long, of a = 45°; By, param, lat. =37° 55'. But measuring 

 them according to their geodesic distance, the equatorial radius 

 A being taken =1, we have 



B«=^" =-78540, 



4 



B 7 =Cy =l){E / -E(52 o 5')}=l-21106-'82225=-38881. 



Reverting to the general formulae for s> A, but writing therein 

 A = l, and therefore C = vl--8 2 ; writing also (£ = 90° (that is, 

 making the formulae to refer to the node N of the geodesic line), 

 we have 



a/1 -S 2 sin 2 / 



^\{n + ^)Xl t (n,c)-c^ fi \; 



A=^- 82 



wcos 



but for the calculation of the second of these formulae by means 

 of Legendre's Tables it is necessary to express Ti^n, c) in terms 

 of the functions E, F. 



The proper formula is given in Fonct. Ellipt. vol. i. p. 137 ; 

 viz. this is 



A f j6 \ n,(n 9 c) = W+ — A A{b, QVfi + Vfii'fa 6) 

 sm0cos0 ,v ' cose/ v ' ' * ' K ' ' 



-V/M(b,'0)-'EfiE(b,6) 9 



where A(b, 6)=\/l — b 2 sin 2 6. 6 is an angle given by the equa- 

 tion cot 6 = \/n; we have w=tan 2 / ; ; consequently #=90° — /'. 

 Substituting this value, except that for shortness I retain 

 E(6, 0), (F(J, 6) in place of E(6, 90°-/'); W» 90°- J'), we have 



A(^6>) = v / l-^cos 2 / / 



= x/l-{I-&sm*l)cos*l' = sin J ; 

 and thence 



tan 0A(Z>, 0) = cot /sin /= - . • 



whence 

 ~, . sin/'cos/'f, , „, r cos/ ,.„ As ^ /7 .."I 



-E /C E(M)}. 



But 



n + c 2 = tan 2 /' + £ 2 sin 2 /= sin 2 / sec 2 / . 

 Hence 



(n + c 2 )IIj(w, c) -c 2 F iC = sin 2 / {sec 2 VTJfa, c) -h*¥f} ; 



