Ball — Contributions to the Tlicori/ of Screws. 



43 



Suppose that (1), (2), (3) be three given screws defining the system, then 

 equation (39) is one of the conditions that (4) shall also belong to the 

 system. The conditions will be expressed by 



when A is the pitch operator. 



If (1), (2), (3) be three coreciprocal screws, then 



r„ = 0, 



0, ^31 = 0: 



and representing (4) by 0, we have the equation fT" = in the form 



1 1,1., 



Ih Pz Ih 



(40) 



If as usual we denote by 0i, 02, di the three coordinates of B with respect 

 to the three coreciprocals, then* 



pfii = '^le, Ihfii = ^19, Ih^z = ^.<e, and ^'e =i^i^i' + l^f^i' + Ih'L', 



which is the well-known expressionf for the pitch of a screw of a o-system 

 expressed in terms of its coordinates. Similar remarks may be made with 

 respect to freedom of the fourth and fifth orders. If six screws belong to 

 a system of the fifth order, there is then only a single condition to be 

 satisfied, which may be written 



U ^ 



ro-j, 2}-. 



•zzr,., w". 



K I 



h ^-.u ^2S. ^'ai 



Wj I ^i2 SJ", 



13 lU 



•sj'ii 3^11 



'ifsi ^Si Wm W, 



51 2^' 



(41) 



Wg[ Wgj OT53 w^ -arg^ 2h 



In this case, as there is only one condition to be satisfied, the formulae 

 A?7= 0, A'U - 0, &c., can be only identities if Z7 = 0. The determinant 

 here given is already known in a different form as the sexiantj of the six 

 screws, and a quaternion expression of the same function is given further on 

 in the present paper (equation 93). 



" Treatise," p. 34. 



t Ibid., p. 3G. 



tlbid., pp. 37, 248. 



