6 REPORT 1866. 



equations by the iiiTuntioii of an algebra of non-commutative symbols. To take 

 the simple instance given by Ellis. 



Noah = the father of Sheni ; 

 . •. A son of Noah = a sou of the father of Sheui = Shem. 



The process is formally identical with the following : — 



Examples of inference lying beyond the domain of the old logic are deserving 

 of much greater attention than they have hitherto received. Professor De Morgan 

 seems to be the only writer who has treated of such examples with any degree of 

 fulness and ability. ' (See papers on the Syllogism, and on the Logic of Relations, 

 in the Cambridge Philosophical Transactions.) 



On Com2)lexes of the Second Order By Dr. Pluckee, F.R.S., of Bonn. 



Dr. Pliicker showed a series of models executed with great accuracy by ]\Ir. Ep- 

 kens of Bonn, calculated to illustrate his theory ot complexes of the second degree. 

 Such complexes are determined aualj-tically by equations of the second degree 

 between the coordinates of right lines'in space. In any plane whatever the lines 

 of such a complex envelope a curve of the second class, and every point in space is 

 the centre of a cone of the second order generated by lines of the complex. If _a 

 plane revolves round any line within it regarded as an axis, the variable conic 

 therein generates a surface. The same surface is enveloped by a variable cone of 

 the second order, the centre of which moves along the same axis. Surfaces of this 

 description are of the fom'th order and the fourth class. The axis is a double line 

 of the surface. The four circumscribed cones whose centres are the four intersec- 

 tions of the double line with the surface, degenerate into systems of two planes, each 

 of which touches the surface along a curve of the second order. In each of four 

 planes passing through the double line, the conic degenerates into two points ; these 

 points (singidar points of the surface) are the centres of cones formed by tangents 

 to the surface. The poles of the double line, with regard to all conies in planes 

 passing through it, are situated on a right line, through which pass the polar planes 

 of the^'doubleline Avith regard to all circumscribed cones. 



The surfaces even of the more general description are easily constinicted ; the 

 models exhibited belong to the special case where the double line is at an infinite 

 distance. In this case the surfaces are formed by curves of the second class in 

 parallel planes, having their centres on a light "line. The circumscribed cones 

 become circumscribed cylinders. 



On the NimcreU'qjtic Functions, Gopel and Wcierstrass's Si/stems. 

 Bii W. H. L. Ettssell, A.B., F.B.S. 

 The author of these papers gave an explanation of the methods discovered by 

 Giipel and Rosenhain for the comparison of the hyperelliptic functions. After 

 pointing out their enormous complication, he stated that a simpler method had 

 been discovered by Dr. Weierstrass, which he illustrated by showing how Abel's 

 theorem had been employed by that mathematician in deducing the periods of 

 elliptic and hyperelliptic functions. 



On a Property of Surfaces of the Second Order. By H. J. S. Smith, F.B.S. 



On the large Piime Number calculated hy Mr. Barratt Davis. 

 %H. J. S. Smith, i^.i?.>Sr. 



On a Nomenclature for Multiples and Suhmdtiples to vernier absolute Stand- 

 ards convenient in practice, and on the fmulamental Unit of Mass. By G. J. 

 Stohet. 



i 



