528 - REPORT— 1902. 



7. Matrix Notation in the Theory of Screws.^ 

 By R. W. H. T. Hudson, lif.A. 



In this paper it is shown how the six coordinates of a screw may be arranged 

 in the form of a four-rowed skew matrix, and that for many purposes the screw 

 and also the correspouding infinitesimal linear point transformation may be 

 represented by this matrix. Instances of this representation are given, and it 

 is shown that the matrix PQ-QP represents a screw whose axes (in the language 

 of elliptic geometry) cut those of the screws P and Q at right angles, and is con- 

 nected with the disjjlacement resulting from a succession of half-turns about P 

 and Q in case these represent lines. The Peterson-Morley theorem, that the 

 common normals of opposite sides of a rectangular skew hexagon have a common 

 normal, is reduced to an identity among these matrices which corresponds to 

 Jacobi's identity in the theory of continuous groups. The notation is useful in 

 condensing the formulae obtained in extending Darboux's theory of rotations 

 depending on two parameters m, r, the chief results being expressed by the 

 equation 



where Vdn + i^dv is the matrix of the instantaneous screw. 



8. On Pluperfect Rumbers. By Lieut. -Colonel Allan Cunningham, R.E. 



If /N denote the sum of the divisors of N (including bolh 1 and N), than N 

 is called Tluperfect if /N -f- N-^^ (an integ<^r >2). A table of 85 even Pluper- 

 fects (P) is presented, of which (i6 (computed by the author) are believed to be 

 new. 



A simple (somewhat tentative) method of evolving them has been found, viz. 

 P = 2''-'. M,, . F, where M.^ = (2' - 1), and the form of F is suggested by M,^. One 

 or more Pluperfects have been found for every value of q up to 39, except 33, 

 •IS, 36; also for y = 45, 51,62. Some are very large numbers, containing more 

 than twenty different primes: the largest is 



P = 2*^'. M,, . F, where M, = (2" - 1) . 3 . -"'+i 



F = 3' . 5' . 7- . 11 . 13 . 19^ . 23 . 59 . 71 . 79 . 127 . 157 . 379 . 757 . 43,.331 . 3,0.33,169. 



No odd Pluperfect has been found. Among even Pluperfects— 



1. None has been found of order ^>6. 



2. Ouly one of order /x = 3 has been found to arise from any one base (2'^-^). 



3. From any one base (2''-i) only two different orders (^, fx') of Pluperfects 

 have been found, viz. either (3, 4) (4, 5) (5, 6). 



Two kinds of simple relations are found to exist between the orders (/x,., (x',) of 

 two sets of P where P, : P',. is constant, viz. 



If Pr = 2»,-'- X, ./, P'r = 2'V-\ X, ./', where /, /. and all the X,'s are odd, 



^/ P/ ■ ■ ■ P/ /' 

 thenalso ^=J;^= . . . ./;>|;„where v = ff^f,.'=ff^f, 



provided/ and/' are prime to every X,.. 



Hence (1) if v = v', then every /x, = ^', ; (2) if /Xj = /x, = &c. = /li/., then /x'i=m''- 



= &C. = 11.', : 



' Messenger of Mathematics, vol. xxxii. p, 51. 



