Mr. A. Cayley on the Theory of Elliptic Motion. 437 



Suppose that q^, Pq, r^, d^, u^ correspond to the time t^ {q is 

 constant, so that go=q), and write 



V=na%u— UQ + e sin u — e sin Uq). 

 Joining to this the equations 



r=a{l—ecosu), rQ=a{l — ecosuQ), 



0-00= tan-' { ^^T^sinu \ _t,n-i( ^r=7 ^sinz<o\ 

 ^ cosM— e / V cosM — e / 



u, Uq, e will be functions of a, r, r^, 0, 0q, and consequently {n 

 being throughout considered as a function of a) V will be a 

 function of a, r, r^, 0, 0^. The function V so expressed as a 

 function of a, r, r^, 0, 0^ is, in fact, the characteristic function of 

 SirW. R. Hamilton, and according to his theory we ought to 

 have 



dV=z^n^a{t—tQ) da +pdr + qd0 —podr^ — q^0Q. 

 To verify this, I form the equation 

 <^V=inffl(u — Mg + esinM— esinMo)</e 



+ na\{\ + e cos u)du—{i +e cos u^du^ 

 + na^ (sin M — sin Mq) de 



+ r \ar—[l —e cos ujda — ae sin udu + a cos ude\ 



1— ecosM*- ' J 



e sin Uq 



•e cos Uq 



Tide sin u 

 , —{drQ—{\—ecosu^da—aesinuQduQ-\-acosuQde\ 



Ue- 



-'^0+ ./T^(l-eco3u^ [(l-.')<^»„+smM.]}. 

 The coefficient of du on the right-hand side is 



g,, , wa^e-sin^M n«^(l — e^) 



na-'fl 4- e cos m) — , ■ :r — ^ '- 



V — ecosu 1 — ecosM 



„/ 1 — e^ + fc^sin^wX 



= na'^\ 1 +ecosM T I, 



\ 1— ecosM / 



which vanishes, and similarly the coefficient of ^/m^ also vanishes : 



the coefficient of de is the difference of two parts, the first of 



which is 



o . na^e sin u cos u nop- sin u 



u(r sm u + 



1 — ecosM 1 — ecos« 



a ■ /t 1— ecoswX 



=na*8inwll— = I, 



\ 1— ecosM/ 



which vanishes, and the secondpart in like manner also vanishes; 

 2F2 



