262 CELESTIAL MECHANICS. 1. vii. Q8. 



[if we multiply p.r'' — gz" by r, and py" — ?'z" by q, we have 

 prx" — qrz"=:0 and pqy"'—qrz":=iO; whence, by subtrac- 

 tion,] qy"—rx"—0. Thus the evanescence of the three 

 fluxions is reduced to the two conditions px"'-=.qz", and 

 py"-=irz'\ which belong to a right line, forming angles with 



3i^\ y"y and z" of which the cosines are ,, „ — ^ r, , 



;, and ,. o o ^ ; consequently this line 



-Jip'^ + q^ + r^^y v/(pM-5' + ^-) 



is at rest, and forms the true momentary axis of rotation, 



since the equations hold good equally with respect to all 

 its points, whatever may be the actual magnitude of their 

 coordinates x\ y', and z'. 



354. Theorem. Retaining the notation 

 of article 345, the angular velocity of rotation 

 is V (p'+^'+r^). 



We may consider the motion of ^ point so situated, 

 that zf^ may be 1, x"—0, and 2/"=^; we shall then have 

 the velocity of this point, in the directions of oc', y'y and 2', 

 by dividing the respective fluxions by d^, and we shall thus 



obtam— — sin 0, — — cos 5, and-- — sm d respectively; con- 

 d^ ' d^ ' d^ ^ ^ 



sequently the whole velocity of the point in question will 



<p — d9 cos 9, and rdt—d-^ sin Q cos ^ + dfi sin (p]. Now 

 dividing this velocity by the distance of the point in ques- 

 tion from the momentary axis of rotation, which is evi- 

 dently the sine of the angle made by that axis with z\ of 



which ■■■ , ' -^ — r — ^. is the cosine, that is, by J ■ ■ * ' ' -, 



I 



