Motion and Spherical Trigonometry. 739 



Then if Pj and Qj come to P and Q when P and Q are 

 starting together from A, when 



u = 0, Ux=w, and ADQ = y = am ?r, cos AEP = Ay, 



(1) S = S = S° = 5x= C03AD ^= C03 ?> D ^=cQo; 



« ^A = ^A = ^=EA ==COsAEPo = A ^ E ^ = EP c 



(3) AQ = ADsiii(/>, DQ = ADcos<£, 



AP = AE./esm<£ = ABsin</>, EP = AE . A</>, 



so that, by Euclid VI. C, 



(4) DX = DQ A p Ql = AD cos (/> cos f, 



AY = A( ^^Q 1 = AD sin <f> sin -f , 



AP AP 



(5) Al r ! = A £ = AE . * 2 sin <£ sin ^ 



= AD sin <£ sin yjr = AY, 

 EX 1 = JE] ^|^ 1 =AE.Ac/>A^. 



_DW DX XW DX AY XW 



(o) cos 7 - AD - AD + AJ) - - AD + - AD • AY ' 



XW XT Dc QqTq QqL_ 

 AY ~ TY ~ eA ~ T A ~ LA ~~ 7 ' 

 (7) cos 7 = cos <£ cos yjr -f sin </> sin ijr A7. 



«n a - EW ' !X t , XiW t EX t AY, X t W , 

 l»J *Y- AE _ AE + AE _ AE~ + AE " AY! ' 



, Q , X,W, X,R Eo P R /DXo 



(9) AY7 = RY7 = ^A=R^: = V M= cos ^' 



(10) Ay = A(f>Ayjr -\- k 2 sin sin i/r cos 7. 



Here (7) and (10) are the w^ell-known formulas of Legendre; 

 and thence, as before in (7) and (4), § 8, 



(11) cos 7 = cos <p cos t/t 4- sin <£ sin -v/r(A(£Ai/r + k 2 sin $ sin i/r cos 7) 



_ cos (f) cos yjr + sin <£ sin yjrA6A\jr 

 ' " 1 — /c'^ sin 2 </> sin 2 1/r 



(12) A7 = A(j)Ayfr + « 2 sin </> sin i/r(cos (/> cos -^ + sin <fc sin ^A7), 

 A _ AcpAyfr + « 2 sin sin ^ cos </> cos i/r 



' 1 — k* sin 2 <£ sin 2 n/r 



3C 2 



