974 



SCIENCE. 



[N. S. Vol. XIV. No. 364. 



and integrating, 



(4) ip^ = A,Wgh{E— cos ^). 



To make the equations homogeneous, put 



(5) 0G = 6, 00=^6\ 4:AiWgh = k% 

 so that 



(6) p2 = JF(^— C0S1?). 



Now denoting the perpendicular GK from O 

 on the tangent at H by p, 



(7) ^ sin i5^ = d'' — (J cos 1^, 

 so that, eliminating i5^, 



and this is the characteristic geometrical prop- 

 erty of a Poinsot herpolhode ; it also defines the 

 ti'ace of a rolling line of curvature, the intersec- 

 tion of an ellipsoid and a confocal hyperboloid 

 of two sheets, and then (5 is the angle between 

 the generating lines of the confocal hyperboloid * 

 of one sheet through H (Darboux). 



Since 

 (9) Km=QH^—GK\ 



A 



(10) 



(Q' — (?cosi9)2 



Wgh_ 



2n^Z, 

 {G' — GzY 



and putting cos ■& = z, 



<"' (S) 



(12) Z= (£_.)(.-., ^^^^^^ 



Denoting the roots of Z= by 2,, z^, Zt, and 

 arranging them so that 



(13) 



•x, ^* -x- dz ^. . „. . 



then with -^r positive, — negative, as m Fig. 1, 



(14) 



Zl > 1 > ^2 > 2 > Zg > — 1, 



dz 

 h V2Z' 



an elliptic integral of the first kind ; and, by 

 inversion, z is an elliptic function of t. 



To make the reduction to Legendre's stand- 

 ard form, put 



*Proc. London Math. Society, XXVI., XXVII. 



(15) 

 (16) 



(17) 

 (18) 



(19) 

 then 



(20) 



z = Z2 sin^ <l> -\- Z3 cos^ i>, 

 z — Zz={z^—z^) sin2 0, 



02 — 2 = (02 — 23) C0S2 ^, 

 01 — 2 = (^l — 23) A2 0, 



A-.2 = ' 



\¥^=-^ 



nt: 



V 



\2i 23/ J^ 



= /C-;^J(^-^rt. 



nt. 



(21) Fc^ = K—mt, mt^/(^-l—±'\ 



Then, in Jacobi's notation, 



(22) <f> = am{K — mt), 

 and in Gudermann's notation, 



(23) z = z^Sn^^iK — mt) + z^CnHK— mt), 



the expression of z or cos ■& by elliptic functions 

 oft. 



Next, denoting the angle KGHhy x, 



d& 



(24) 



tan X- 



GK GK sin ^ 



V{'2A^WiihZ) 



V2Z 



6'— J' 



(25) sin ^ cos x = 



G^— G, 



sin 1? sin x 



V \2AT^Wyh{E — z)'] ' 



Vz 



V {E—z) 



Resolving transversely to GH, or rather, tak- 

 ing the moment of the velocity of H round G, 



dtz 

 (26) P^-^T = Og ■ GK= WgU&in d- ■ GK 



= WgJi {G^ — G cos ^) 

 so that, from (6), 



(27) 



dTT 



G' — G^ 



W—21.+ 



dt 2Ai{E — z) 

 G^ — GE 



2A 



S^—6E 



GET dt 



T 



iE — 2) y'2Z 



involving an elliptic integral of the III. kind, 

 and then 



(29) ■4, = Tc~-x. 



