swinging through an Arc of Finite Magnitude. 855 



the quantity usually denoted by K. Let the elliptic integral 

 corresponding to BAC be denoted by u 9 then the elliptic 

 integral corresponding to ABC may be denoted by K — u. 

 Thus 



angle BAC = am u, 



angle ABC - am(K— u). 



The formulae concerning the angles and sides of the right- 

 angled triangle ABC can now be translated into the language 



of elliptic functions. For instance 



i 

 tan BAC X tan ABC = 



cos AB 



But BAC = am u, ABC = am(K-w) 



cos AB= yl — /r, 



so that tan u tan (K— u)= 7< .: • 



v l — K- 



Other formulae may be obtained in a similar manner. 



Theorem in Spherical Trigonometry. 



We now proceed to prove a theorem in Spherical Trigono- 

 metry which will be found useful hereafter in studying the 

 Addition-theorem. 



If on the arc AB of a great circle there be erected two 



triangles ACB, ADB, having the angles at C and D each 

 equal to a right angle, and if perpendiculars AE and Bb be 

 let fall from A and B respectively onto the great circle 

 passing through the vertices C and D, then 



(i.) CF = DE, 



(ii.) OAB= DAE, 



sin(AE + BF) 



(iii.) sin fADE.+ BCF) = 



sin 



AB 



