VELOCITIES, &c. WITH RESPECT TO AXES MOVEABLE IN SPACE. 17 



also, since p', ff are involved in precisely the same manner as p, 6, it follows that 



where a, /3', 7' are what a, /3, 7 become, when p is put for p. 

 From these equations we obtain 



cot' 0' = - -t;^, 



but a' = 1 —7- = 1 - 



a'/3'7' 1 + a/37 cot' 0' 



1 1 _ 1 + /37 cot= e 



If" "" ' ~ Af ■ r+^^7Co?y ~ " 1 + a^7 cot= ' 



whence, with the corresponding expressions for /3 , 7 , 



(1 4- c.i67 cot^ e)' 



cot= Q =- cot^ ^ • (1 ^ ^^ cot-' Q){\ + 7a cot- 0)(I + a^ cot^ 0) ' 



hence p', Q' are known in terms of p, 0. 



32. Substituting now for p , 6' in terms of p, 6, we obtain from equation (8) 



d(cot 0) _ , ,2 cot 

 ~d^ " ^ coit? 



= ± A/^ - (1 +|87cot^0)(l +7acot^0)(l +a/3cot^0)|i, (10) 



and from equation (9) 



Q = hp^ (1 + a)37 cot- 6). 



If A be known by means of (7), these two equations determine completely the motion of 

 01 the axis of angular velocity in altitude and azimuth, since p, and therefore a, j8, 7, are 

 constants. 



If (b denote the azimuth at any instant, -j- = Q, and dividing the last equation by the 

 ' at 



preceding, we obtain a relation involving (p and 9 only, which will therefore be the differential 



equation to the conical path of 01 in space ; and it is worth notice that, this relation being 



independent of h, the path of 01 is the same whether the body be acted on by a couple whose 



axis coincides with OH, or whether it be acted on by no forces. The effect of the couple in 



this case is in fact only to alter the velocities of the different lines, not the paths which 



they describe. 



Also equation (l) gives to = hp^ sec 6, from which w is known when 6 is known by means 



of equation (10), and thus the velocity about 01 is known completely as well as its position at 



any time. 



33. If there be no forces acting, i. e. if G = 0, h is constant, as is also o) cos 0, the re- 

 solved angular velocity of the body about OH. Also the vis viva of the body 



w 



-' r cos 9 



and is therefore constant; and hence — is constant, or war; both well known results. It may 



r 



Vol. X. Part I. 3 



