518 Proceedings of the Royal Irish Academy. 



XXYI. XoTE Olf A GEOilETEICAL HeTHOD OF IxVESTIGATr5"G THE 



DrxAiincAL PfioPEEXiEs OF THE CrLiXDEoiD. Bt Eobeet S. Ball, 

 LL.D., P.E..S., Ptoyal Astronomer of Ireland. 



[Eead, May 11, 1885.] 



It is a fundamental point in the theory of screws, that when a certain 

 condition is fulfilled between two screws they are ' reciprocal,' i.e. a 

 twist on either does no work against a wrench on the other. Jl a, h 

 be the pitches of the two screws, and if d be their shortest distance 

 apart, and 6 be the angle between them, then the condition that they 

 shaU be reciprocal is thus stated : 



{a + h) cos 6 - d ?,m.6 = 0. 



Let a and /3 be any two screws, then if a body receives a twist 

 about a, followed by another twist about /?, the position arrived at 

 could have been reached by a single twist about a third screw y. If 

 the amplitudes of the twists about a and /3 are given, then the position 

 of y, as weU as the amplitude of the resultant twist thereon, are, of 

 course, both determined. If, however, the amplitudes of the twists 

 on a and yS are made to vary while the screws a and /3 themselves 

 remain fixed, then the position of y, no less than the amplitude of the 

 resultant twist, must both vary. A little reflection will, however, 

 show that the position of y will remain constant so long as the ratio 

 of the amplitudes of the twists about a and (B remains unchanged. 

 As this ratio varies, the position of y will vary, so that this position 

 is a function of a single parameter ; and, accordingly, y must be re- 

 stricted to be one of the generators of a certain ruled sui-face, which 

 includes a and /? as extreme cases in which the ratio is zero and in- 

 finity respectively. 



It is proposed hereiu to investigate the nature of this ruled surface 

 by the Theory of Reciprocal Screws. The real character of this surface, 

 which is called the cylindroid, is of course well known, as it forms a 

 fundamental part of the Theory of Screws. It is only the method 

 of investigation which forms the novelty in this communication. 



Let ^ be a screw which is reciprocal to both a and ^, then it will 

 be obvious that 6 must also be reciprocal to y ; for suppose a wrench 

 on 6, then a body can be twisted about a and y8 without any expendi- 

 tiu'e of work. If the body be restored to its original position by a twist 

 backwards along y, then no work can be done during this operation, 

 for otherwise there would be a quantity of energy created or lost. It 

 must not, however, be supposed that the theorem of the reciprocity 

 of 6 and y is limited only to the case of a system of forces in 

 which the doctrine of the conservation of energy is true. For the 



