10 ■ KEPORT — 1875. 



It is not possible, perhaps, to hasten the arrival of this generalization beyond a 

 certain point ; but we ought not to forget that we can hasten it, and that it is our 

 duty to do so. It depends much on ourselves, our resolution, our earnestness, on 

 the scientific policy we adopt, as well as on the power we may have to devote our- 

 selves to special investigations, whether such an advent shall be realized in our day 

 and generation, or whether it shall be indefinitely postponed. If governments would 

 understand the ultimate material advantages of every step forward in science, how- 

 ever inapplicable each may appear for the moment to the wants or pleasures of 

 ordinaiy life, they would find reasons, patent to the meanest capacities, for bringing 

 the wealth of mind, now lost on the drudgery of common labours, to bear on the 

 search for those wondrous laws which govern every movement, not only of the 

 mighty masses of our system, but of every atom distributed throughout space. 



Mathematics. 



On a Screw -complex of the Second Order, 

 By Professor E. S. Ball, LL.D., F.R.S. 



Denoting by 6^, . . . , B^ the six coordinates of a screw, then an homogeneous 

 equation of the second degree, Ue=0, between the sLx coordinates denotes what 

 may be termed a screw-complex of the second order. If c* be a given screw, then 



being a linear equation in 6^, . .., 6^, denotes the locus of screws about which a 

 body which has freedom of the fifth order can be twisted. To this system one 

 screw (ji is reciprocal ; and we may call the screw 6 thus defined the polar of the 

 screw u, with respect to the screw-complex 11^ = 0. The relation between « and 

 its polar is independent of the screws of reference. 



The locus of the screws about which a body can twist so that when it has the 

 unit of twist velocity its kinetic energy is zero is an imaginary screw-complex of 

 the second order. The polar of any screw » with respect to this screw-complex is 

 the screw an impulsive wrench on which would make the body commence to twist 

 about ». 



On tlie Analytical Form? called Factions. By Professor Catley, F.R.S. 



A faction is a product of difierences such that each letter occurs the same number 

 ot times ; thus we have a quadrifaction where each letter occurs twice, a cubifac- 

 tiou where each letter occurs three times, and so on. A broken faction is one 

 which is a product of factions having no common letter ; thus 



{a-h)~(c-cl)(d-e){e-c) 



is a broken quadrifaction, the product of the quadrifactions 



{a~hf and {c-cl)(d-e){e-c). 



We have, in regard to quadrifactions, the theorem that every quadrifaction is a 

 sum of broken quadrifactions such that each component quadriiaction contains two 

 or else three letters. Thus we have the identity 



2{a-l){h-c){c-d){d-a) = (b-cy .((i-df-{c-ay . {b-d)^+{a-hy .(c-d)*, 



which verifies the theorem in the case of a quadrifaction of four letters ; but the 

 verification even in the next following case of a quadrification of five letters is a 

 matter of some difficulty. 

 The theory is connected with that of the invariants of a system of binary quantics. 



