TRANSACTIONS OF THE SECTIONS. 9 



Any line (s) upon this surface is considered to be a screw, of which the pitch is 



c+acos2^, 

 where c is any constant, and 6 is the angle between s and the axis of x. 



The fundamental property of the cylindroid is thus stated. If any three screws 

 of the surface be taken, and if a body be displaced by being screwed" along each of 

 these screws through a small angle proportional to the sine of the angle between 

 the remaining screws, the body after the last displacement will occupy the same 

 position that it did before the first. 



For the complete determination of the cylindroid and the pitch of all its screws, 

 we must have the quantities a and c. These quantities, as well as the position of 

 the cylindroid in space, are completely determined when two screws of the system 

 are known. 



In the model of the cylindroid which was exhibited, the parameter a is 2'6 inches. 

 The wires which correspond in the model with the generating lines of the surface 

 represent the axes of the screws. The distribution of pitch upon the generating 

 lines is shown by colouring a length of 2-6 x sin 26 inches upon each wire. The di- 

 stinction between positive and negative pitches is indicated by colouring the former 

 red and the latter black. 



It is remarkable that the addition of any constant to all the pitches attributed in 

 the model to the screws does not affect the fundamental property of the cylindroid. 



When a rigid body is found capable of being displaced about a pair of screws, it 

 is necessarily capable of being displaced about every screw on the cylindroid deter- 

 mined by that pair. 



The theorem of the cylindroid includes, as particular cases, the well-known rules 

 for the composition of two displacements parallel to given lines, or of two small ro- 

 tations about intersecting axes. 



If the parameter a be zero, the cylindroid reduces to a plane, and the pitches of 

 all the screws become equal. If the arbitrary constant which expresses the pitch 

 be infinite, we have the theorem for displacements, and if the pitch be zero, we 

 have the theorem for rotations. 



As far as the composition of two displacements is concerned, the plane can only 

 be regarded as a degraded form of the cylindi'oid from which the most essential 

 features have disappeared. 



0*1 the Number of C'ovariants of a Binary Quantic. 

 By Professor Catlet, D.C.L., F.E.S. 

 The author remarked that it had been shown by Prof. Gordan that the number 

 of the covariants of a binary quantic of any order was finite, and, in particular, 

 that the numbers for the quintic and the sextic were 23 and 26 respectively. But 

 the demonstration is a very complicated one, and it can scarcely be doubted that a 

 more simple demonstration will be found. The question in its most simple form is 

 as follows : viz. instead of the covariants we substitute their leading coefficients, 

 each of which is a " seminvariant " satisfying a certain partial differential equa- 

 tion ; say, the quantic is («, S, e. . . -kj^x, y)", then the differential equation is 

 (abb + ^ibc- ■ ■ ■ +»yOit)«< = 0, which qua equation with «4-l variables admits of n 

 independent solutions : for instance, if w = 3, the equation is (ad b + 2&B e-|- 3cB d)u = 0, 

 and the solutions are a, ac — h^, d^d~3abc+2P ; the general value of m is m= any 

 function whatever of the last-mentioned three functions. We have to find the ra- 

 tional non-integral functions of these functions which are rational and integral 

 functions of the coefficients ; such a function is 



i {(«2<?-3a6c+263)2+4(ac-62)3}, 



=a^d'+4a(^+4b^d-3Pc>-6abcd, 



and the original three solutions, together with the last-mentioned function a'^d^ ■+■ Sec, 

 constitute the complete system of the seminvariants of the cubic fimction ; viz. every 

 other seminvariant is a rational and integral function of these. And so in the 

 general case the problem is to complete the series of the n solutions a, ac — 6^, 



