the Elliptic Equation. 205 



.". SH = HP. Now in the right-angled triangle S G H we have 

 the angle 8 = 0), and the side SH=-r + ^ (the latitude of P 

 being \). The hypothenuse SG is therefore found by the equa- 

 tion tan SG . cos o) = tan ( t + h ) • If we call y the latitude of 



a, we have SG= -r- + qj ^^^ therefore 



The arc 7 has two values, depending on the double value of \. 

 It must also be noted that there are two auxiliary circles, accord- 

 ing to whether the tangent circles pass through the north or 

 south poles. One is easily found from the other by putting 

 TT-yS for y3. 



The equation which gives \ is easily found to be (since \ is 

 one side of the triangle of which a and /S are the others, and w 

 is the angle opposite /S) 



cos « sin \ + cos ft) sin a cos X, = cos /?. 

 This gives 



tan { -; — -- I = ^\ \/Um^B — sin^asin^w) +sinacosa> L 



\4 2/ cosa+cos^Sl ^ ^ '- V 



whence 

 tan 



(-7 —%\ = 7^\ v^sin^/S— sin^asin^o)) + sinacosa> >• 

 4 2/ cos« + cos^l ^ '- J 



This form of the equation is very inconvenient. 



Knowing the meridian on which the centre of a circle lies, the 



circle is completely determined by its modulus Q and its mean 



latitude /z., where 



. ^ sin a . cos /9 



smc7=-^ — 5, sm/i,= 



sin/S' cos«' 



and .•. cos Q . tan/i . tan/S=l, tan a = tan ^ cos /a. 



In the auxihary circle, let p be the polar distance of its centre, 

 q its radius, t) its mean latitude, and fits modulus ; then we have 



tani(« + p) = rX -/(sin^/S— sin^a . sin^w) +sina.cos(ol, 



2\i—rj cosa+cospl ^ J 



COS (m) 



.-. tan i (0-1- /v) .tani(o— ») = 7— , Trrgfsin^/S— sin^w) 



^y^ ' ' ' ^^^ ^' (cosa + cosp)''^ ' 



= C03^a).tan^(/9 + a) .tan ^(/9— «); 



