776 PEOFESSOE 8TLVESTEE ON THE MOTION OF A EIGID BODY 



Calling the two parameters ~jj > -jj > j„ j,* respectively, an inspection of the system 



of equations (1, 2, 3, 4) at pp. 767, 768 will show that the angles a, /3, y, |, >j, J^, u are 



M 

 known, and may be registered in a table when jj t, j,, q, are given, the time t being 



reckoned from some determinate epoch, which must be so fixed as to be identical for 

 the disk and the associated bodyf. We may assume as such epoch indifferently the 

 moment when the axis of the disk has its maximum, or when it has its minimum incli- 

 nation to the invariable line, i. e. when the quantity (cos yf in the equations 



(cosa)='+ (cos/3)^+(cosy)='=l,1 



(cosa)^ (cos^)^ (cosy)' _ > -^ 



* Calling A, B, C the original moments of inertia, it is important to notice that we have seen that no real 

 distinction of motion arises from =, lying between — and - on the one hand, or between — and — on the other ; 



the so-called two kinds of polhods and Legendbe's primary distinction of the problem into his cases (1) and (2) 

 turn entirely upon this difference, but the two kinds of motion are convertible into one another (save as to the 

 correction for the uniform displacement round the invariable line) by the theory of contra-relation. The real 

 essential distinction of cases can only arise from particular values being assumed by q^, q^. 



The quantities 0, 5^, q^, 1, q^+q^ are written in natural ascending order. 



The two singular cases are (A) when qi=q^, which is the case of two equal moments, (B) when q^=l, which 

 is Legendbe's ' Troisiome Cas,' ' Cas trfis-remarquable," arts. 26, 27, corresponding to the instantaneous axis 

 describing the so-called " separating polhod." 



Besides these properly called singular cases, there are what may be termed special cases arising from sequences 

 of two or of three terms in the above quinary scale becoming approximately equal, or subequal, in Mr. De 

 MoBGAw's language, which relation may be denoted by the ordinary sign of equivalence. 



Thus we shall have special cases when 



q^^O, OT q^=q„ OT l = q^, OT q^+q^=l, 

 and double-special cases when 



5,=g,=0, 1 = ?,=9'„ q^+q^=l = q,. 



The last of these is of course tantamount tol^^^q^ with q^^O. But even this table does not exhaust all the 

 specially notable cases ; for in the first of the double-special cases which corresponds to that of a body differing 



little from a sphere, we may again mark off as extra-double special the ease where il ^ 0, and also that where 



2i=l. 



It does not fall in with the plan of this paper to investigate these several cases, but they are probably all 

 of them deserving of particular examination. 



t "We may express the motion in terms of the parameters g',, y, as foUows, writing a?, y, z for cos «, cos /?, cos y : 



ffi 9, 9i + 9. 



(1) 



dz_ 

 df 



,,=,^.,4^_giJ.. (3) 



