﻿186 Sir G. Greenhill on 



6. Next to make the axle rise to a cusp on d = 2 from 

 = 6 Z in the penultimate pseudo-regular precession, the 

 impulse applied is KK 3 , to make the axle come to rest on 

 the horizontal G 2 K 3 in K 2 , C 2 , in fig. 3. 



Then in the general formula, or on the figure, with CD 

 the perpendicular on K 3 M 3 , 



G 2 K 3 2 -G 2 K 2 2 = OK 3 2 -OK 2 2 = CK 3 2 -:P 2 (cos6> 2 -cos^ 

 or CK 3 2 = K 3 D . K 3 M 3 =2 OM ,MK^, 



and producing MK double length to K', 



K 3 M 3 . 0C=20M . MK = OM . MK', 



implying that if OCLN' is the parallelogram on OC, the 

 diagonal OL will cut MK' in L 3 such that ML 3 = M 3 K 3 , and 

 K 3 is determined by drawing L 3 K 3 parallel to 00, cutting 

 off the length CK 3 on OK, in fig. 3. 



7. If the impulse is applied about the axle of the top, to 

 increase 00 to 00 3 , and make the axle rise to a cusp in 

 fig. 3, 



OC 8 .K 3 M 3 = 20M.MK, with MK = K 8 M„ 



so that OC 3 = 20M. 



Thus the axle will rise from the horizontal in fig. 1 to a 

 cusp by the application of an axial impulse CC 3 = 0C. 



8. The impulse might be applied about a vertical axis to 

 the steady motion, making K rise vertically to K 3 ; and 

 then in a cusped motion, with 00 changed to 0C 3 , and rising 

 to 0C 2 at K 2 on the level of G 3 K 3 in fig. 4 ? 



G 3 K 3 2 -G 3 K> = OK 3 2 -OK 2 2 = C 3 K 3 2 =MK 3 . K 3 D 



= 20M . MK(cos0 2 -cos0 3 ) = 2OM . MK^ D , 



MK 8 . OG 3 = 20M . MK = 0M . MK', 



MK 3 _ MC 3 _ OM 

 MK'~M0'~00 3 ' 



dropping the perpendiculars K 3 3 , K'C on 00 ; and drawing 

 the circle on the diameter OC, with ordinate MQ', 



C 3 M . C 3 = 0M . M0' = MQ' 2 . 



