swinging through an Arc of Finite Magnitude. 857 

 Hence 



sin(ADE + BCF) 



= sin ADE cos BCF + sin BCF cos ADE 



sin DBA . . _ -r^ sin CAB . ^^ . „ 



sin AE cos BF + . nii sin BF cos AE 



sin AD ' sin BC 



_ sin AE cos BF + sin BF cos AE 

 ~ sin AB 



_ sin(AE+ BF) 

 sinAB 



The Addition-theorem. 



In order to prove the Addition-theorem, we may employ 

 Legendre's investigation on spherical triangles, adapting it 

 to our present system of interpretation. 



Suppose that AB is the minor axis of a sphero-conic o£ 

 the nature already discussed. Let and D be two 

 points on the sphero-conic lying on the same side of AB ; 

 join CD, CA, CB, DA, DB. Drop the perpendiculars AE 

 and BF on CD produced, and bisect CD in the point (Jr. 

 Draw the polar circles of A and B ; let them intersect AB 

 produced in H and K respectively, and let L denote that 

 point of intersection of the two circles which lies on the 

 same side of AB as C and D do. Produce AD, BC to meet 

 LH, LK in M and N respectively. 



Suppose next that the great circle joining C to D suffers a 

 slight displacement, in such a way that the area of the 

 segment cut off from the sphero-conic remains unaltered, 



Phil. Man. S. 6. Vol. 19. JSo. 114. June 1910. 3 K 



