442 Proceedings of the Royal Irish Acadetny. 



sible in one way, while still retaining six co-reciprocals for the screws 

 of reference. 



The pitch pa. of the screw a is given by the equation {Screws, 

 p. 35) 



Pa=Pia.i~ + . . . + PqO^^: 



the six screws of reference being co-reciprocals, the function pa. must 

 retain the same form after the transformation of the axes. The dis- 

 criminant of the function 



U+Xpa. 



equated to zero will give six values of X ; these values of X will de- 

 termine the coefficients of Um the required form. I do not, however, 

 enter further into the discussion of this question, which belongs to the 

 genei'al theory of linear transformations. 



The transformation having been effected, an important result is 

 immediately deduced. Let the transformed equation be denoted by 



(11) ai^H- . . . + (66)a6- = 0, 

 then we have 



A = ^^(66)ae; 



whence it appears that the six screws of reference are the common 

 screws of the two systems. We thus find that in this special case of 

 tomography 



The six common scretvs of the two systems are co-reciprocal. 



It is proved (Scretvs, p. 48) that the correspondence between im- 

 pulsive screws and instantaneous screws is of the type here referred 

 to. The six common screws of the two systems are therefore what we 

 have called the j^rincipal screws of inertia, and they are co-reciprocal. 



The special circumstances under which a screw a has the same cor- 

 respondent, whichever of the two systems a be regarded as belonging 

 to, demands a few words. If we take the six common screws of the 

 two systems as the screws of reference, then the condition stated can 

 only be fulfilled when the relation has the form 



A = ± «^i» 

 A = ± as. 



