305] THE GEOMETRICAL THEORY. 325 



304. An Important Exception. 



If p a = 0, then (a?;) is 90, and consequently p a tan (a?;) is indefinite. 

 If, therefore, the pitch of the instantaneous screw be zero, then we are no 

 longer entitled to locate the centre of gravity in a certain ray. All we know 

 is that it lies in the plane through a perpendicular to 77. In general the 

 knowledge of the impulsive screw corresponding to a given instantaneous 

 screw implies five data, yet this ceases to be the case if p a is zero, for as 77 

 must then be perpendicular to a there are really only four independent data 

 given when 77 is given. We have, therefore, in this case one element the less 

 towards the determination of the rigid body. 



305. Two Pairs of Impulsive and Instantaneous Screws. 



Let us next suppose that we are given a second pair of corresponding 

 impulsive and instantaneous screws. We shall examine how much further 

 we are enabled to proceed by the help of this additional information towards 

 the complete determination of the rigid body in its abstract form. Any data 

 in excess of nine, if not actually impossible, would be superfluous. If, 

 therefore, we are given a second pair of impulsive and instantaneous screws, 

 the five data which they bring cannot be wholly independent of the five data 

 brought by the preceding pair. It is therefore plain that the quartet of 

 screws forming two pairs of corresponding impulsive screws and instantaneous 

 screws cannot be chosen arbitrarily. They must submit to at least one 

 purely geometrical condition, so that the number of data independent of each 

 other shall not exceed nine. 



It is, however, not so obvious, though it is certainly true, as we found in 

 281, that the two pairs of screws must conform not merely to one, but to 

 no less than two geometrical conditions. In fact, if 77, be two impulsive 

 screws, and if a, /3 be the two corresponding instantaneous screws, then, 

 when the body acted upon is perfectly free, the following two formulae must 

 be satisfied : 



P Pf* 



cos + cos 



cos (a??) cos 



We can enunciate two geometrical properties of the two pairs of screws, 

 which are equivalent to the conditions expressed by these equations. 



In the first place, each of the pairs of screws determines a diameter of 

 the momental ellipsoid. The fact that the two diameters, so found, must 

 intersect each other, is obviously one geometrical condition imposed on the 

 system a, 77 and /3, . 



