ACTED ON BY NO EXTfiENAL FOECES. 775 



the time with the squared velocity. In the case of disk motion there is a distinctive 

 feature which is deserving of notice. In this case we have 



Hence 



AB(cu1+a>l)=(A-{-B)M.-V, 



showing that the angular velocity with which the disk turns about a line in its own plane 

 is constant throughout the motion, whilst the velocity about the axis perpendicular to its 

 plane is continually varying, in the first particular agreeing with, and in the second dif- . 

 feiing from what takes place fox a body of three dimensions with two of its principal 

 moments of inertia equal. 



It is easy to see how in the general case every conceivable motion of a body of any 

 form may be tabulated and reduced to a table of treble entry, and how greatly the use of 

 such tables may be facilitated, and seemingly distinct cases reduced to identity by aid of 

 the twofold method of reduction above explained. Let us consider the case of a body 

 whose principal moments of inertia are A, B, C, arranged in ascending order of magnitude. 



We have seen that the quantity p must always be intermediate between ^ and g- 



If the direct reduction be employed instead of x' «' p' rs' ^® ^^^^^ ^^''^^ 



1 1 1 M _L- ±- J_- Mi- 



A~^' B"'"' C~^' IJ~^'^^^Ai' B,' C,' W 



and if jj is intermediate between -g and ^, ji will be intermediate between g- and q-, 



where C,=A,+B,. 



On the other hand, if the supplemental method be employed, 



% 1 1 1 ^ 1 1 M „^„ 1 1 1 M' 



^~a' ''"b' ''"c' ^-p'^yc^' B'' r' P' 



where C'= A' + B' will take the place of -r > =r> 7=; so that if ir^ is intermediate between 



"^ ABC L* 



■jr and g' Ya will be intennediate between -gy and q, • 



L*. 



Hence by using the direct method of reduction in the case where j^ is greater than B, 



L* 



and the supplemental method of reduction where rr is less than B, the original body can 



be always replaced by a disk of which A„ B„ A.+B, are the new principal moments of 

 inertia, L the given initial impulsive couple, M the new vis viva, and where the ascending 



order of the magnitudes is -^^ , ?4 ' ^ ' ^> so that 5¥ ' t¥ will be both of them less 

 ^= A + B L* B A L* L* 



than unity. This reduction being effected when the motion of the disk is known, that 



of the associated body is given. 



5n2 



