Motion and Spherical Trigonometry . 735 



8. The addition formula of the elliptic function follows at 

 once from a Legendre spherical triangle, in the manner 

 employed by Mr. Kummell (' Analyst/ 1878) ; for in the 

 spherical triangle AQY, 



(1) sinp= sin QY= sin AQ sin QAY = k sin $ sin yfr', 

 cosj9= v^(l — /rsin 2 </> sin 2 -*//), 



fo>i w cosAQ A<f> 



(2) cos AY = -^ = -^-, 



cos QY cosp 



. A v sin AQ cos QAY _ k sin </> cos yfr' 



sin A 1 _ -~= — , 



cos Q 1 cos p 



cosQV Ayjr' 

 (3) cos Y V = Z v = — L —, 



cos Ul COS V 



QY cos// 



. __ sin QV cos QV Y k sin \lr' cos d> 



sm YV= t^tt = r , 



cos QY cosp 



(4) cos c = cos AY = cos (AY + Y V) 



A^Ayjr' — ac 2 sin <f)COS(f) sin ijr' costy' 

 1 — k 2 sin 2 sin 2 -v// 



= dn (u + K — Mj) s= dn (K — w) — dn (K + w?) . 

 In a similar manner, with Spherical Trigonometry, 



(5) c0S AQY= s i , ^ A ^AQ = s HW^, 

 v ' cos QY cosj> 



• aav cos QAY cos t// 

 sm AQ Y = j^r- = — ? 



cos Q 1 cos p 



sinQVYcosQY sin </>A^' 



(6) cosYQY = — 7=y^ = — r — — , 



v y ^ cos QY cosp 



• ttav cosQYY cos$ 

 sm VQY = - — 7=y&- = r> 



^ cosQY cos^' 



(7) cos AQY= cos (AQY + YQY) 



_ sin cf)A(f) sin -^rAty' — cos (f> cos ^ 



1 — /e 2 sin 2 </> sin 2 -*// 

 = — cn{u-\-K — u l ) = — cn(K — w) = cn(K + w) 9 



(8) AQV= am(K + ir), DQV=i7r + am(K-w). 



9. In the reciprocal diagram of fig. 4, drawn on the left 

 hand, R is the pole of XY, and the perimeter of the triangle 



