﻿Pseudo- Reg alar Precession. 183 



or with cos = :, and replacing OG, 00 for dynamical 

 homogeneity by Darboux's 2A/j, 2AA', 



^y=2;i 2 (F-^(l-^)-4(A 2 -2/Ji'cos(9+A' 2 )==2n 2 Z, 



thus defining z as an elliptic function of t. 

 Resolved into factors, we write 



r L — z x — z. z 2 — z. z — z s , 



in the sequence oo >c x > 1 >z 2 >z> c 3 > — 1 ; and then 



z = z 2 sn 2 \mt + z* 6 en 2 ^mt, m 2 = 2n 2 (^ 1 — %). 



4. Here with 00 horizontal in fig. 1, cos #3=0, 



OK 3 2 -00 2 2 =G 3 K s 2 -G 3 K 2 2 =CK 3 2 



= 200 . OK. cos0 2 = 2CK, . OK, CK 3 = 20K. 



A greater impulse would make the cusps open out into 

 loops in the pseudo-regular precession; but the cusps would 

 be blunted into waves if the impulse was reduced. 



Reverse this tap, and K is brought back again to 0, and 

 the axle would fall as at first from a cusp and rise again. 



In the first cusp motion where the axle rises to a series of 

 cusps and sinks again to the horizontal, the motion is found 

 to be pseudo-elliptic and can be expressed in a finite form, 



sin 6 exp (yjr — lit) i 



= \/(l — cos# ? cos#) +i >/(cos # 2 eos# — cos 2 0), 



connecting azimuth i/r with 0, the inclination to the zenith. 

 The verification is left as an exercise. Here h = h' cos# 2 . 



In the second cusp motion, where the axle is horizontal 

 and falls from a cusp, and then sinks down to an angle 6$ 

 with the downward vertical, nadir, the (yjr, 0) motion is not 

 pseudo-elliptic ; but azimuth yjr and hour angle <f> change 

 place (0, (j>, ijr the Eulerian angles), and 



sin 6 exp (cft — h't) i= V (sec 6 3 — cos 3 . cos 6) 



+ zV (cos # 3 — cos 6 . cos # + sec # 3 ), 



OR 

 where Darboux's h'= ^— = i/(^ . sec S — cos 6 Z ) changes 



place with h, or d' with ~d ; h = h' cos # 2 and B now zero. 



But an interchange again of </>, ijr will give the (\jr, 6) 

 motion of a non-spinning gyroscopic wheel, or spherical 



