15a 1,1, — Conlributlons Iv the Thcorij of Screws. 17 



memoir* it became necessary to introduce the Theory of Screw co-ordinates, the 

 virtual cocfHcieut as a function of the two screws, and now no longer zero, 

 assumed a significance which it had not previously appeared to possess. 



It may perhaps be thought strange that after the lapse of so many yearn 

 it should now have been found necessary to re-examine the rigour of thi' 

 original expression for the virtual coefficient. That expression was given in 

 terms of the pitches j)^, p^ of the two screws o and j3, of d the shortest distance 

 between their two axes, and of the angle between them, and was statedf to 

 be 



i{{l>a + I'p) cos f) - dsin B\. 



In the course of the Quaternion developments of the Theory of Screws, 

 to which a considerable part of the present paper is devoted, a doubt arose, 

 not indeed as to the formal accuracy of this expression, but as to the rigour 

 of the process by which it was supposed to have been established. It presently 

 appeared that there was a flaw in the proof, owing to the absence of any 

 definite convention as to the way in which the angle between the two sci-ews 

 is to be measured. If the angle 360° - 9 had been used instead of 0, then 

 the second term in the virtual coefficient would have a positive sign instead 

 of a negative sign ; and so far as the original deduction of the expression was 

 concerned there was nothing to show which of the two angles was to be 

 used in the expression of the virtual coefficient. The ordinary rule for 

 estimating the angle between two vectors does, no doubt, distinguish between 

 and 180" - 6. It fails, however, to distinguish between and 360° - 0. 

 Unless, therefore, some further convention be established, the virtual co- 

 efficient must have an ambiguous sign for its second term. Our immediate 

 object is to establish the convention necessary so that in all cases the sign 

 of the second term shall be negative. Fortunately it is possible to establish 

 such a convention in the case of two vector-screws which do not intersect. 



I had already attempted^ to remove this uncertainty in the mode of 

 specifying the angle between two screws; but as the result was not com- 

 pletely satisfactory, I have returned to the subject. I am glad to say 

 that the difficulty has now been overcome, and a great improvement in 

 the foundations of the Theory of Screws is the result. I here set down the 

 method of obtaining the virtual coefficient in the way 1 would desire it to be 

 obtained if I were commencing to write the Theory of Screws over again. 



To the apparatus of the Theory of Screws as it lias hitherto existed the 

 important addition of the geometrical conception known as the vector-screw 



* Trans. Hoy. Ir. Aciui., vol. xxv., \i\k 259-327 (1S74). 



t " Treatise," p. 17. 



; Trans. Koy. Ir. Acad., vol. xxxii., ]>\i. lO'J-11,) (1902). 



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