382 



NATURE 



[February i6, 1893 



a point upon a nerve terminal, and that this terminal will be 

 most excited by the light. At the end of the paper Dr. Stanley 

 Hall's views of nerve structure are examined. Captain Abney 

 thought the results of the zoetrope experiments were what one 

 would have expected when pigmentary colours were used. To 

 be conclusive, such experiments must be conducted with pure 

 spectrum colours. The statement about the size of star 

 images being less than that of a nerve terminal would pro- 

 bably need revision. Speaking of colour vision, he said the 

 modern view was to regard light as producing chemical action 

 in the retina, which action gave rise to the sensation of colour. 

 On the author's theory he could not see how colour-blindness 

 could be explained. Mr. Trotter said he understood Helm- 

 holtz to have proved that nerves could distinguish quantity, but 

 not the quality of a stimulus. Since the speed at which stimuli 

 travelled to the brain was about 30 metres a second, the wave 

 length of a light vibration, if transmitted in this way, would be 

 very small. Taking Lord Kelvin's estimate of the minimum 

 size of molecules of matter, it followed that there must be many 

 wave lengths in the length of a single molecule. This, he 

 thought, hardly seemed possible. Mr. Lovibond pointed out 

 that the observations referred to by the author could be equally 

 well explained on the supposition that six colour sensations 

 existed. The confusion of colours he had mentioned arose from 

 lack of light. Mr. Stanley replied to some of the points raised 

 by Captain Abney. In proposing a vote of thanks to Mr. 

 Stanley, the chairman said it had been shown that light could 

 be resolved into three sensations, but it was not known how this 

 resolution occurred. Prof. S. P. Thompson said the gist of Mr. 

 Stanley's paper seemed to be that lights of different colours 

 were concentrated at points situated at different depths in the 

 retina, the violet falling on the part nearest the crystalline lens, 

 and the red furthest away. Another view of the action was 

 that the different sensations might be due to the vibrations of 

 longer wave length having to travel greater distances along the 

 nerve terminals before they were completely absorbed. 



Mathematical Society, January 12. — Mr. A. B. Kempe, 

 F.R.S., President, in the chair.— The President (Prof. Elliott, 

 F.R. S., Vice-President, in the chair) read a paper on the appli- 

 cation of Clifford's graphs to ordinary binary quantics (second 

 part). In the first part it was pointed out that by some small 

 modifications and a recognition of the fact that the covariants 

 of /(j;, y) are invariants of the two quantics /(X, Y) and 

 (X^ — Yx), the theory of graphs, which had been left in an un- 

 finished state by the late Prof. Clifford, furnished a complete 

 method of graphically representing the invariants (and therefore 

 the covariants) of binary quantics. The method as modified 

 depends essentially on the fact that any invariant, when multi- 

 plied by a suitable number of polar elements U,U',V,V', &c., 

 can be expressed as a "pure compound form" (or sum of two 

 or more such forms), the product of a number of "simple 

 forms." Each of the latter has a "mark," viz. one of the 



letters a, 6, c, and has also a certain valence, o, i, 2, 3, &c. 



and these being given it is fully defined, e.g., the simple form of 

 mark a and valence 3 is graphically 



having three radiating bonds, and is algebraically 



aoUVW4-«i(U'VW + UV'W-hUVW') + ao(UV'\V' 

 + U'VW' + U'V'W) + a3U'V'W',' 



the pairs of polar elements U,U'; V,V'; and W,W', corres- 

 ponding to the three bonds of the graphical representation. A 

 pure compound form is graphically represented by a number of 

 simple forms having their bonds connected so that there are no 

 free ends. If in the algebraical expression of a compound form 

 two simple forms both contain the pair of polar elements U,U', 

 there will be a bond connecting their graphical representations ; 

 if the two simple forms both contain two pairs of such elements, 

 viz. U,U' and V,V', there will be two bonds connecting their 

 graphical representations and so on ; if they contain no common 

 pair their graphical representations will have bond connecting 

 them. A pair of polar elements will appear in two simple 

 forms only, so that each bond in the graphical representation of 

 a compound form corresponds to a distinct pair of polar ele- 

 ments. If the algebraical expression corresponding to a graph 

 be multiplied out, it will be found to consist of two distinct 

 factors, viz. :— (i) the product of all the polar elements, and (2) 



NO. 1216, VOL. 47] 



afunctionof the letters «„, aj, a, ; 6(„l>i,l>2, ; &c., &c. ; 



corresponding to the marksa, 3, &c. of the simple forms 



contained in the compound form represented by the graph, the 

 latter factor being an invariant of the quantics 



(«o. «i. ^2< ««)(•*■«>'> . 



&c., &c. 

 where a is the valence of the simple forms of mark a, which are 

 here supposed to be all of the same valence, and similarly in the 

 case of iS, 7, &c. 



In this second part a method of algebraically representing 

 invariants is considered, which is directly derivable from the 

 method of the first part, and was suggested by the graphs ; but 

 differs essentially from the earlier method in that it is indepen- 

 dent of the use of polar elements. It shows, moreover, that the 

 graphs may be regarded as absolutely equivalent to the invari- 

 ants they represent, in lieu of being equivalent to those invari- 

 ants multiplied by a number of polar elements. This second 

 method deals in the first instance with "primary" invariants, 

 i.e. invariants of two or more quantics linear in the coefficients 

 of each. If these quantics are 



(ao. ^1, ^2, aa){x,yY 



(^)> ^1. h bp){x,y)$ 



Sec, &c., 

 and we take 



a = a-i---+ao— — 'r i^^-t— + &c. ad infinitum 

 duQ dUi da^ 



^ = diy— + d2 —r + ^3 — ■ + &c. ad infinitum, 

 db^ doy db^ 



Sec, &c 



we may express any primary invariant by an expression, or the 

 sum of two or more expressions, consisting of the product of 



differences of the operators a, b, operating upon the product 



of the corresponding leading terms, a,,, b^^, &c. Thus 



(a - bfa^bi^a.^^ - 2a^b-^ + aj)n 

 is an invariant of the two quantics 



a^x^ -f 2a^xy + a^y"^, 

 b^x' + 2b-^xy -f b.^y'', 

 linear in the coefficients of each ; and 



{a -b)\a - c)a„b,yC^;^asboCQ - a.J}^c^ - ia^b^c^ + 2a^b.,c^ + 

 «iVo-«oVi 

 is a similar invariant of the three quantics 



t Uf^x^ + yi'^x'^y + sajjy'^ + a^y^ 



bf,x^ + 2b-^xy + b.2y^ 



CaX^-c^y. 



These two invariants are graphically represented by 



® © and ® ® © 



respectively, where the relation between the algebraical and 

 graphical expressions is obvious, viz. to every letter / in the 

 algebraical representation there corresponds a nucleus including 

 the mark /, and to every factor {p-q) in the algebraical represen- 

 tation there corresponds a bond connecting the nuclei of marks 

 / and q. 



We can pass to invariants of higher degrees in the coefficients 

 of the various quantics by substituting like coefficients for unlike. 

 Thus, if we make b^ = a^, b-^ = a^, b.-, = ac,, the primary invariant 



a.J!)^^ — 2a-J)^+afjb2, 

 becomes the invariant of degree 2 



2(aoa2-ai2) 

 of the single quantic 



0^"^+ 2a^xy -t- a^v"^. 



This invariant will be graphically represented by substituting 

 the mark a for the mark b in the graph representing the corre- 

 sponding primary invariant. 



If we proceed to deal in the same way with the invariant, 

 •a'sVo ~ ^2^o<^i ~ 2ao^ifo + 2aJ>^c^ -f- a^b^^ - a^b^^, 

 we get, as the invariant represented by substituting for the marks 

 b and c the mark a, the expression of the third degree. 



