292 Prof. Sylvester on Periodical Changes of Orbit, fyc. 



Hence 



dP + dg*-2co S g>.df.dff 



* ' (sin ft)) 2 



Let 



ff '+/=*> g-f= v - 



Then, by trigonometry, 



Hence 



1 + COS ft) = —^-P j 1 — COS ft) — 



<i _ (du) 2 (l — cos &)) — (<fo) 8 (l + cos co) 

 W ~ (sirT^) 2 



(c 8 -t; g )rfM a + (c g -f< a )<fo 8 > 



and again, 



/ • q, _ /{ds) 2 —(dff _ df— d g . cos co _ du{\ — cos&)) — <fo(l •+ cosft)) 

 V (ds) 2 sin ft) c?5 2 sin co ds 



_ (c 2 -v q )du-{u 2 -c q )dv 

 4<fg sin co ds 



_ ( c g-p«)Ai + _(tt«-c*)-|fe _ (c 2 -t; 2 )^-(c 2 -^ 2 )^ 



4/(7 sin ft) ds ^(w 2 -z; 2 )((c 2 -2) 2 )^4- (c 2 ~tt 2 )^)' 



and similarly, 



(c 2 — v 2 ) dw 4- (c 2 — w 2 ) dv 

 sm^ = 



It is worthy of passing observation that the above expressions 

 lead immediately to the integral of the fundamental equation in 

 the addition of elliptic functions; for if we callj», q the two per- 

 pendiculars from the foci upon ds, we have 



±- '- '- i i — _ = 4ifq sin 6 , sin ?? = 4pq. 



Q(c^-v 2 )du 2 -(u 2 -c 2 )dv 2 ) JJ FH 



Suppose now 



4pq = c 2 — a* 2 . 



be found in Balzer, and probably elsewhere, as affirmed concerning the 

 four sides of a wry quadrilateral ; for any four lines in space which meet in 

 a point being given, a wry quadrilateral may be formed with sides parallel 

 respectively to the same. The- above equation enables us to express the 

 element of a curve in space in terms of vectorial coordinates and their dif- 

 ferentials. 



