666 



Procrc'dings of the Royal Irish Academy. 



and, therefore, if A be reciprocal to the first screw of reference, the 

 formiila to be proved is 



Xn_i ^ 2X'p— =0. 



A few words will be necessary on the geometrical signification of 

 the differentiation inyolved. Suppose a Dyname X to be referred to 

 six co-ordinate screws of absolute generality, and let us suppose that 

 one of these co-ordinates, for instance Xi, be permitted to vary, the 

 corresponding situation of X^ also changes, and consideriag each one of 

 the co-ordinates in succession, we thus have sis routes established along 

 which A will travel in correspondence with the growth of the appro- 

 priate co-ordinate. Each route is, of course, a ruled surface ; but the 

 conception of a mere surface is not adequate to express the route. We 

 must also associate a linear magnitude with each generator of the sur- 

 face, which is to denote the pitch of the corresponding screw. Taking 

 A and another screw on one of the routes, we can draw a cylindroid 

 through these two screws. It will now be proved that this cylindroid 

 is itself the locus in which a moves, when the co-ordinate correlated 

 thereto changes its value. Let 6 be the screw arising from an increase 

 in the co-ordinate Aj ; a wrench on 6 of intensity 0" has components of 

 intensities 6"i, . . . 0"e. A wrench on A has components A"i . . . A'V 

 But from the nature of the case. 



0". _ 6"z 

 A".! A s 



A" 



If therefore 6" be suitably chosen, we can make each of these ratios 

 - 1, so that when 6" and A" are each resolved along the six screws of 

 reference, all the components except 6"i - X"i shall neutralize. But this 



can only be possible if the first reference screw lie on the cylindroid 

 containing 9 and A. Hence we deduce the result that each of the six 

 cylindroids must pass through the corresponding screw of reference ; 

 and thus we have a complete identification view of the route travelled 



