406 Prof. Encke on the Calculation 



For the chords we shall obtain these expressions : 



(12) = 4sini(E. 2 -E l )* {a* sinf (E 2 + E^'-f & c 

 (13)* 4sin|(E 3 -E 1 )-* {a' sin* (E^E^ fr* cos 



(4) (1 4) * = 4 sin * ( E 4- E i)' { a * sin * (E 4 +E!)H- ft* cosj (E 4 +E,)*} 

 * = -' ' * ' * 



(23)* = 4sini(E 3 -E 4 )' {a'sini (E 3 +Eo)*+ 6' 



(24) 4 -=4sinf (E 4 -E 2 )' {a^sinj (E 4 +E 4 )*+ 6 cos$ (E 4 +E 4 )} 



(34)* =4sinf(E 4 -E 3 )4 {a^sini(E 4 -HE 3 ) 2 -f 6^ cos J (E 4 +E 3 )*} 



to which many other different forms, more convenient for in- 

 dividual cases, may be given. 



The four equations (2) being multiplied together, and the 

 square root being taken of the product, we obtain 



V (1 23) (1 24) (134) (234) 



= 16 a* b* \ sin * ( E 2- E i) sin * ( E 3-Ei) sin i (E^EJ) 

 ? sin 4 (E 3 -E 2 ) sin \ (E 4 -E 2 ) sini (E 4 -E 3 )$ 



an equation in which all six combinations of the equations 

 occur. The product of any two of the equations (2) gives 

 only five combinations 



l 

 J 



Dividing the above square root by each of these products, 

 we obtain the following equations : 



(124) (234) m sini(E 4 -E 2 ) 

 (123) (134) : sinltEj, EO 

 (5) /(I 3 4) (2 3 4) _ sini(E 4 -E 3 ) 



(123) (124) 



(124) (134) S inj(E 4 -E 1 ) 

 (123) (234) 



Assuming now 



(123) (124) 



v/ 



N/ 



(123) (134) 

 (1|24) (234) 



(123) (234) 



(124) (134) 

 we have 



tanrr (45 + t ) - 



' 432 1 



tang 



