OF ARTS AND SCIENCES : FEBRUARY 8, 1870. 203 



and putting 



3 — g X °- 



Then the equations become 



r sin (a — <r) = a_ v 



r sin (« -|- S) = a , 



r sin (J -{- <r) = a x . 

 Or, 



«i 4- a -i 



r sin g> cos o- = -=— ! -. 



2 



r sin (co -|- 5) = « , 



ai — a _ i 

 r cos or sin o- = -. 



If r sin » and r cos <a are eliminated from these equations, and we 

 make 



«i + a-i 



2 a 



Ql — <* — l 



- cos 8 = a = c cos £, 

 sin S = b = c sin /3, 



2a 



where c may be taken as positive and the quadrant of /3 becomes deter- 

 minate, or /3 may be assumed between the limits ± 90°, there will be 

 obtained, for the determination of <r, the equation 



sin 2 a- = 2 c sin (a- -\- /3). 

 The computation of c and /3 may be facilitated by introducing the 

 auxiliary quantities k and (, such that 



k sin f = -p — 



V2a 



£ cos £ = -rj— , 



V2a 

 then 



c cos /3 = & cos (4o° — f) cos S. 



c sin /3 = £ sin (45° — £) sin 8. 

 It is evident that the determination of o- depends on the solution of 

 an equation of the fourth degree ; but its value can be very readily 

 obtained from the above equation by the tentative process ; and then 

 r and o> by means of the equations 



aok cos (45° — Q 



r sin &> 



r cos a 



cos a- 

 a k sin (45° — £) 



sin a- 

 and finally xo and rj by means of the relations given above. 



