40 THE THEORY OF SCREWS. [44- 



It p a be indefinitely great with respect to a and d a i, then 



p a cosf(ai ) pa cos (oci) 



-- 2a &quot;2a 



_ ;j a cos (as) 

 &quot; 



26 &quot; &quot; 4 ~ 26 



_ Pa cos (as) _ Pa cos (as) 



a 5 ~ 2c 2c 



If the co-ordinates of a screw not itself at infinity satisfy 



ai + a, = 0; a :j + a 4 = 0; 5 + a 6 = 0; 

 then we must have 



for the equations 



(Pa + ) cos (ai) - dai sin (ai) 



ttj = & &amp;gt; 



_ (p a a) cos (ai) - d al sin (ai) 



and two similar pairs could not be otherwise satisfied. 



We are not however entitled to assume the converse, i.e. that if the pitch 

 is infinite then the three equations a 1 + a. 2 = 0, &c. must be satisfied. It will 

 however be true that 



but some at least of the co-ordinates being infinite, we are in general 

 prevented from replacing these equations by the ordinary linear form. 



45. Indeterminate Screw. 



It may however be instructive to investigate otherwise the circumstances 

 of a screw a possessing the property that its six co-ordinates a 1( a, ... a^ are 

 submitted to the three conditions 



! + a, = ; a s + a 4 = ; a 5 + 6 = 0. 



Two distinct cases must be considered. Either the screw a must have some 

 finite points, or it must lie altogether at infinity. The first alternative is now 

 supposed. The second will be discussed in the next article. 



If there be any finite points on a then for such points the three left- 

 hand members of the equations in 43 are all zero. The three right-hand 

 members must also reduce to zero. The only way in which this can be 

 accomplished (for we need not consider the case in which all the co-ordinates 

 are zero) is by making p a infinite. 



