June 5, 1890J 



NATURE 



127 



not suited for testing uneducated persons. A similar instrunaent, 

 introduced by Chihret and Meyer, of Paris, is to be found in 

 ophthalmic hospitals. 



I may further remark that I do not consider any test satisfac- 

 tory unless made by an ophthalmic surgeon, as he alone is 

 accustomed to deal with such people every day of the week, 

 and can alone eliminate such errors as refraction-disease and 

 stupidity. D. D. Redmond. 



14 Harcourt Street, Dublin, May 3. 



The Green Flash at Sunset. 



Your correspondents (vol. xli. pp. 495, 538) seem to imply that 

 this phenomenon is only seen at sea, but I observed it on May 

 17 while walking from east to west, near Worms Heath (VVar- 

 lingham, Surrey). It had been an exceptionally fine day, since 

 the morning, and about 8 p.m. there was not a cloud in the sky, 

 except to westward, where strips of cloud were rapidly forming, 

 and covering up the glow of sunset ; the sun had sunk behind a 

 hill, when, suddenly, my companion and I both saw a flash of 

 green light against the thickest cloud ; it lasted I or 2 seconds, 

 just long enough for there to be no doubt about it. We com- 

 pared it to the glare thrown by "green fire," extending over an 

 area whose diameter appeared about four times that of the moon. 



At 12 p.m. the same night it was raining. 



I think this observation definitely negatives the sea-wave 

 theory, while the appearance was seen at least in association 

 with the condensation of aqueous vapour. Perhaps the reason 

 it was not <J/«/j/j -green was that this vapour absorbed the blue 

 rays? T. Archibald DuKES. 



16 Wellesley Road, Croydon, June 2. 



THE THEORY OF SCREWS} 



THE book before us, a large octavo volume of over 

 600 pa^es, gives in a connected form the results of 

 Sir R. S. Ball's investigation in the theory of screws, as 

 contained in his "Theory of Screws'' and a series of 

 publications in the Proceedings and Transactions of the 

 Royal Irish Academy. But as its scheme is that of a 

 texi-book on theoretical mechanics, it begins with a chap- 

 ter on the postulates and methods of mechanics ; whilst 

 chapter vii. is on the theory of moments of inertia; chapter 

 viii. on impulsive forces capable of imparting to a rigid 

 body a given state of velocity ; and chapter x., on kinetic 

 energy, contains a number of propositions from analytical 

 dynamics. Here expressions for the kinetic energy, for 

 its change in consequence of an impulse, Lagrange's 

 equations of motion in generalized co-ordinates, Hamilton's 

 principle of least action, and various other propositions, 

 are developed in the usual form — that is to say, without the 

 use of screws. The rest of the book relates to the theory 

 of screws and its applications. This alone, as forming 

 the characteristic feature of the book, concerns us here, 

 and of it we shall try to give an outline. 



In order not to be unintelligible to those who have no 

 knowledge of Ball's creation, it will be necessary to begin 

 with the very elements of the subject ; and in order to 

 form a just idea of the scope and importance of the new 

 method, it will not be sufficient to give a sketch of the 

 results obtained — it will be necessary to take a wider 

 view of the subject. We shall then be able to form some 

 idea of the inherent capabilities of the theory. These I 

 believe to be very great — very great indeed. One of its 

 peculiarities lies in this, that all the results obtained in 

 modern algebra and geometry, as distinct from analysis, 

 seem to be directly applicable to it. 



Friends of synthetic geometry and of graphical methods, 

 too, will find here a wide field for investigations. Grass- 

 mann's " Ausdehnungslehre " has already been pressed 

 into its service, and the theory of vectors and quaternions 



' " Theoretische Mechanik starrer Systeme auf Grund der .Methoden und 

 Arbehen, und mit einetn Vorworte von Sir Robert Ball, Royal Astronomer 

 of Ireland." Herausgegeben von Harry Gravelius. (Berlin : Gearg Keimer, 

 1889.) 



NO. 



1075, VOL. 42] 



is easily applicable. Clififord, in fact, has generalized 

 the latter theory into that of biquaternions to embrace 

 screws. 



Mr. Cartesius, to make use of Sir Robert's personifica- 

 tions, has been dethroned, and Mr. Anharmonic together 

 with Mr. One-to-one reign in his place. 



Poinsot, whose investigations form the starting-point of 

 the theory of screws, has proved that a rigid body can 

 always be transferred from one position to any other by a 

 rotation about a certain perfectly determined axis, to- 

 gether with a translation along this axis. These two 

 motions combine to a motion identical with that of a nut 

 on a screw. It is completely determined if the angle 

 through which the rotation takes place, together with the 

 ratio of the translation to the rotation, is given. This 

 I ratio — the '■^ pitch " oj the j'^r^w— characterizes the screw. 

 j As the motion does not at all depend upon the diameter of 

 j the screw, we may suppose this to become infinitely small, 

 and then we have the notion of Sir R. Ball's screw. 



A screw, therefore, is a line in space which has con- 

 nected with it a certain pitch— /.^. a certain length, as the 

 pilch is a linear magnitude. The compound motion con- 

 sidered is called a '• twist " about a screw, and is known 

 if the screw and the " amplitude " of the twist, i.e. the 

 amount of rotation, is given. In the same way a system 

 of forces can, according to Poinsot, always be reduced, 

 and that in one way only, to a single resultant and a 

 couple turning about the resultant ; and these two dis- 

 similar parts Ball combines to a " wrench on a screw," 

 the line of action of the resultant force being the axis of 

 the screw and the ratio of the moment of the couple to 

 the force giving the " pitch," whilst the magnitude of the 

 resultant force is called the '' intensity " of the wrench. 



We have thus a new entity — the screw— and its intro- 

 duction forms the characteristic distinction of the theory. 

 Connected with it is a kinematical and a kinetic entity — 

 the twist about a screw, and the wrench on a screw. 



If we now consider a rigid body under the action of any 

 forces, then the latter combine at every moment to a 

 wrench on some screw, whilst the motion itself is always a 

 twist about some other screw. If the body is constrained 

 in any manner, then the reactions due to the constraint 

 will also at every moment combine to a wrench about 

 some screw. 



The problem first to be solved is that of the com- 

 bination of twists and wrenches. Let any two screws, a 

 and iS, be given, then wrenches on them constitute together 

 a system of forces, and therefore combine to a wrench 

 on some other screw, 7, which has to be determined. If 

 the ratio of the intensities of the two given screws be 

 varied whilst the screws themselves remain unaltered, 

 then the screw, y, of the resultant wrench also varies, and 

 its axis describes a surface called the cylindroid. This is 

 a ruled cubic surface which can be described as follows : — 

 Let through a fixed line, /, a plane be drawn, and in it a 

 circle be taken. Let a point, P, move uniformly in the 

 circumference, whilst the plane itself turns uniformly 

 about /, completing half a revolution whilst P describes 

 the whole circumference. The perpendicular from P to / 

 will then generate the cylindroid, and the screw on any 

 generator will have a pitch equal to the length of the 

 perpendicular from P to /. The line / is a nodal line of 

 the surface and perpendicular to all screws on it. All 

 cylindroids are similar, and through any two screws one 

 cylindroid can always be drawn. The projections of all 

 generators on a plane, perpendicular to the nodal line, 

 form a flat pencil in which each ray corresponds to one 

 screw. Also to each point on the circle corresponds one 

 screw. We may here mention that this generation of the 

 cylindroid stands in a very close relation to the plane re- 

 presentation of the cylindroid which is given in chapter 

 XX. For if A, B are the ends of the diameter of the 

 above circle which is perpendicular to the nodal line /, 

 then to A and B correspond two generators of the cylin- 



