RIGID BODY ABOUT A FIXED POINT. 293 



This, taken in conjunction with (37), gives 



Sag +■ Sea = 0, 



whose integral is 



Sag = constant . = — Q cos /3 , suppose, . . . (39). 



Equation (36) may now be written 



B£ = -(A-B)Qacos/3, 

 or 



Bs = — (A — B)Qu cos j8 + constant vector. 



But we have always, by (14), (see § 24) 



p = y , 

 or by, (35), (36), (39), 



Ba + (A-B)aQcos/3 = y .... (40). 



So that the constant vector is y . 



Thus we see that a and s are always coplanar with y, and that each remains 

 constantly at the same inclination to it. 



47. Operating on (40) byS.s, S.a, S . 7, respectively, we have 



-BQ 2 -(A-B)Q 2 cos 2 /3 = -A 2 , 

 -BQcosj8-(A-B)Qcos/3 = Say, 



- BA 2 + (A - B)Sa 7 Q cos 13 = 7 2 , 

 and these give, in order, 



(Acos 2 /3 + Bsin 2 /3)Q 2 = h 2 , 



- AQ COS /S = Say, 



- (A 2 cos 2 jS + B 2 sin 2 /3) Q 2 = y 2 . 



The first and third determine /3 and in terms of the given constants h 

 and T7, and the second gives the value of the constant inclination of a to the 

 fixed line 7. 



Introducing — a 2 , which is unity, as a multiplier of y 2 in the third equation, 

 and adding to its members the squares of the corresponding members of the 

 second, we have 



-B 2 Q 2 sin 2 /3 =N 2 a 7 . 



48. We get equations immediately derivable from these by seeking at once 

 the equations of the fixed and rolling cones, by which the motion may be exhi- 



I bited. Thus the locus of s in the body, i.e., the rolling cone, has by (14) and (38) 

 the equation 



QT ?S = TyTe , 



VOL. XXV. PART II. 4 F 



