BIBLIOGRAPHICAL NOTES. 511 



except all rays save those contiguous to any one ray then the congruency may be 

 regarded as linear. Hence any property of a linear congruency must apply to 

 the Hamiltonian system as restricted in the proposition before us. 



A linear congruency is constituted by those screws of zero pitch whose 

 coordinates satisfy two linear equations. They are the screws which belong to 

 a 4-system and which each have zero pitch ( 76). But we know that each such 

 screw must intersect both of the screws of zero pitch on the cylindroid reciprocal 

 to the 4-system ( 212). It has also been shown that any transversal meeting two 

 screws of equal pitch on a cylindroid must intersect at right angles a third screw 

 on that surface ( 22). Hence the shortest distance from any ray of the congruency 

 to the axis of the cylindroid must lie on a generator of the cylindroid. This is 

 however only true for one particular ray. 



Hamilton s most instructive theorem shows, more generally, that the shortest 

 distances between any specified ray R of the congruency and all the other contiguous 

 rays have a conoidal cubic as their locus, such as might be represented by the 

 equation 



z (x 2 + y 2 ) = A y? + 2Bxy + Cy 2 . 



There are two disposable quantities in the selection of the origin and the axis of x. 

 If these quantities be so taken as to render A ~ ; (7 = 0, then the equation is at 

 once shown to represent a cylindroid of which R is the axis. Of course all rays 

 of this congruency intersect two fixed rays, and the axis of the cylindroid must 

 also intersect both of these rays. 



MOBIUS (A. F.) Lehrbuch der Statik (Leipzig, 1837). 



This book is, we learn from the preface, one of the numerous productions to 

 which the labours of Poinsot gave rise. The first part, pp. 1355, discusses the 

 laws of equilibrium of forces, which act upon a single rigid body. The second 

 part, pp 1 313, discusses the equilibrium of forces acting upon several rigid 

 bodies connected together. The characteristic feature of the book is its great 

 generality. I here enunciate some of the principal theorems. 



If a number of forces acting upon a free rigid body be in equilibrium, and if 

 a straight line of arbitrary length and position be assumed, then the algebraic sum 

 of the tetrahedra, of which the straight line and each of the forces in succession 

 are pairs of opposite edges, is equal to zero (p. 94). 



If four forces are in equilibrium they must be generators of the same hyper- 

 boloid (p. 177). 



If five forces be in equilibrium they must intersect two common straight lines 

 (p. 179). 



If the lines of action of five forces be given, then a certain plane S through 

 any point P is determined. If the five forces can be equilibrated by one force 

 through P, then this one force must lie in S (p. 180). 



To adopt the notation of Professor Cayley, we denote by 12 the perpendicular 

 distance between two lines 1, 2, multiplied into the sine of the angle between them 

 (Comptes Rendus, Vol. Ixi., pp. 829-830 (1865)). Mobius shows (p. 189) that if 

 forces along four lines 1, 2, 3, 4 equilibrate, the intensities of these forces are 

 proportional to 



^23. 24.34, 713714734, ^12. 14.24, ^12. 13.23. 



It is also shown that the product of the forces on 1 and 2, multiplied by 12, 

 is equal to the product of the forces on 3 and 4 multiplied by 34. He hence 

 deduces Chasles theorem (Liouville s Journal, 1st Ser., Vol. xii., p. 222 (1847)), 

 that the volume of the tetrahedron formed by two of the forces is equal to that 

 formed by the remaining two. 



