ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 117 



COS * I 



U(ia) e^u 



ilijia Hti ' 



cos <t> = 



and so substituting and reducing, we obtain the following expressions for 

 the nine cosines : — 



_ 0,(0){n(u + ia) + ll(u — ia)} 

 " ~ 2U^{ia)e{u) ' 



Q^{0){l\{u-iria)-\l{u—ia)} 

 2i Hi(m) e(w) ' 



e(m) H^rt 



a = 



a =• 



/3' = 

 /?" = 



r = 



7 = 



tt 



7 = 



H,{ia) e(M)' 



e(0){n^(u-hia)+llXu—i(t)} 

 2Hj(«0 0tt ' 



2iHi(i«) Git * 



II,(za) Gm' 



H^(Q){9(M+m) — 9(^—1(0} 

 2JHjnf On ' 



Hi(0){ e(it + ?«) + e (u—{a) } 



2Hj(ta) ew ' 



II(<'a ) BiH 

 i'H^ia 9m 



These functions can, of course, be expanded in scries by the formulae given 

 in section 10, Part II. of this Iteport. Jacobi in his memoir enters into a 

 discussion of the ambiguities occasioned by the use of the symbol t= V — 1, 

 which I omit here, my object being to give a clear insight into the principle 

 of the method by which the problem of the motion of a rigid body round 

 a fixed point is solved. 



Section 3. — In the 50th volume of Crelle's Journal there is a memoir by 

 Lottner on the motion of a rigid soHd of revolution round a fixed point 

 which is not its centre of gravity, but which is situated in the axis of revo- 

 lution. This memoir is very similar in its character to Jacobi's. I shall 

 content myself therefore with giving results. 



The equations of motion are given by Poisson in the following form : — ■ 



dd/ 

 Cncos0 — A8in^0-- = Z, 

 at 



(Id) d\L 



~=n + co8d~, 

 at at 



* Tbeee ralues of ccs and cos ^ are of course derived from equationB (3). 



f 



