TRANSACTIONS OF SECTION A. 547 



hind wheels, so that upon running down the side of a hill upon a tramway the 

 apparatus shall preserve a horizontal position. That, being weighted to ascertain 

 the balance, which can only be effected by repeated trial, the truck be liberated. 

 The effect will be to inflate the fabric into a parachute — to elevate the wing-arms 

 and stretch the pectoral cord, and if properly balanced the machine will descend 

 safely in the direction of its head. 



If it be found that a weight equal to that of a man can be brought to the 

 ground safely, then might come the manual test, and his capacity to depress the 

 wing-arms so as to keep up an undulation. 



18. On the desirability of observivy Occultation of Stars, of the first and other 

 bright 'inagnitudes, from places where they are to be seen near the 

 horizon. By H. S. Williams, M.A., F.B.A.8. 



MATHEMATICAL DEPARTMENT. 



1. Sur la representation des rotatiotis autour d'un point par des points de 

 I'espace. By Cyparissos Stephanos. 



2. On the Polar Planes of a point with respect to four Quadric Surfaces. 

 By W. Spottiswoode, M.A., Pres. B.S. 



3. On the Extension of the Theory of Screws to the Dynamics of any ma- 

 terial system. By Robert S. Ball, LL.D., F.B.8., Royal Astronomer- 

 of Ireland. 



A mechanical system of any number of rigid pieces, anyhow connected, is dis- 

 placed to an indefinitely close position. This displacement could have^een pro- 

 duced by giving to each piece a twist of definite amplitude about a given screw. 

 We thus have a series of pi-imary screws equal in number to the pieces of the 

 system. If the twists on any two consecutive screws be compounded they will 

 form a twist on a third screw, which we may speak of as being inter^nediate. The 

 whole series of screws primary and intermediate is called a screw-chain and, com- 

 bined with a single amplitude or metric element, expresses a displacement of 

 the system. 



If the system have but one degree of freedom, then the only possible movements 

 are those of twisting on one screw-chain. If the system be capable of displace- 

 ment about a second screw-chain it must necessarily be capable of displacement 

 about a singly infinite number of other chains, which can be simply constructed, 

 when a third chain has been found. The result of the first twist and the second 

 twist could have been directly obtained by a third twist. We thus have three 

 screws corresponding to each element. These sets of three are each co-cylindroidal. 

 We thus have a number of cylindroids equal to the number of elements of the 

 system. To find another screw-chain about which the system can twist, select any 

 screw on one of the cylindroids, and a homographic screw on each of the others ; 

 this will give the primary screws. Each set of three intermediate screws are also 

 co-cylindroidal, and the required intermediate screws are also a homographic 

 system, 



lu the case of a system capable of twisting alx)ut three arbitrary screw-chains, 

 it can be shown that the determination of subsequent chains can be obtained by 

 the aid of point homogi-aphies in planes. With freedom of the fourth order the 

 point homographies of spaces may be used. For liigher orders of freedom concep- 

 tions analogous to those of 'Parallel Projections' in statics are available. 



N V 2 



