Ball — Contrihutions to the Theory of Screios. 663 



It will of course be understood that / is perfectly arbitrary, but 

 results of any interest are only to be anticipated when / has been 

 chosen with special relevancy to the Dyname itself, as distinguished 

 from the influence due merely to the screws of reference. We shall 

 first take for /the square of the intensity of the Dyname, the expres- 

 sion for which is found in the Theory of Screivs, p. 34, to be 



^ = af + . . . + Ofi- + 2aia2(12) + . . ., 



where (12) denotes the cosine of the angle between the first and second 

 screws of reference, which are here taken to be perfectly arbitrary. 

 The second group of reference screws we shall take in a special form. 

 They are to be located two by two on three intersecting rectangular 

 axes {Screws, pp. 46, 172) : so that 



i2 = (Ai + Ao)2 + (X3 + A4)- + (/\5 + XeT- 



Introducing these values, we have, as the first emanant, 



:§ai/3i + S(ai^3 + %A)(12) = (/xi +/X3)(Xi + A,) + {fJ.3 + fJ^i)i>^3 + K) 



+ {fM^+lXeXh + K) ; 



but in the latter form the expression obviously denotes the cosine of 

 the angle between a and (3 where the intensities are both unity ; hence, 

 whatever he the screws of reference, we must have for the cosine of the 

 angle between the two screws the result 



5aiySi + S(ai/?2 + ao/5i)(12), 



an expression otherwise arrived at in Trans., E. I. A., vol. xxv., 

 Science, p. 306. 



In general we have the following formula for the cosine of the 

 angle between two Dynames multiplied into the product of their inten- 

 sities : — 



ciUi aug 



This expression, equated to zero, gives the condition that the two 

 Dynames be rectangular. 



If three screws, a, /3, y, be all parallel to the same plane, and if 

 be a screw normal to that plane, then we must have 



dR 



dui 



^ dR 



dR 





dR 

 dy^ 



dyo 



11., VOL. Ill, — .SCIENCE. 



3r 



