302 



Prof. F. Holmgren. 



[Jan. 13, 



On examining the equations resulting from the eliminations of the 

 variables, it turns out that they can be rationally transformed into ex- 

 pressions such as UP' — U'P=0, where U and XT' are qnadrics, and P and 

 P' linear functions of the variables remaining after the eliminations. 

 The forty-eight co-ordinates then consist of the twenty-four coeffi- 

 cients of the four functions of the form U (say the U-co-ordinates), 

 together with the twenty-four coefficients of the functions of the form 

 U' (say the XT'- co-ordinates), arising from the four eliminations respec- 

 tively : viz., 4x6 + 4x6=48. And it will be found that the co- 

 efficients of the forms P, P', are already comprised among those of 

 U, XT' ; so that they do not add to the previous total of forty-eight. 



The number of identical relations established in the present paper 

 is thirty-four. But it will be observed that the equations UP 7 — U'P=0 

 are lineo-linear in the U-co-ordinates and in the U'-co-ordinates ; and 

 as we are concerned with the ratios only of the coefficients, and not 

 with their absolute values, we are, in fact, concerned only with the 

 ratios of the u-co-ordinates inter se, and of the U'-co-ordinates inter se, 

 and not with their absolute values. Hence the number of independent 

 co-ordinates will be reduced to 48 — 34 — 2 = 12, as it should be. 



The thirty-four identical relations arrange themselves firstly in two 

 sets : one set belonging wholly or principally to the U-co-ordinates, 

 and the other set wholly or principally to the U'-co-ordinates. In 

 each set there are four groups : one of four, one of eight, one single, 

 and again one of four equations ; seventeen in all. In the course of 

 the paper, the two groups of eight are obtained in two forms : first, 

 by a purely algebraical method in a rational form ; and secondly by a 

 method partly geometrical and partly algebraical, in an irrational 

 form. 



II. " How do the Colour-blind See the different Colours ? In- 

 troductory Remarks." By Frithiof Holmgren, Professor 

 of Physiology, University, Upsala. Communicated by 

 W. Pole, Mus. Doc, F.R.S. Received December 6, 1880. 



That the colour-blind do not see colours in the same way with 

 normal-eyed persons we may know from the fact that they confuse 

 rays of objective light which, to the normal eye, give quite different 

 impressions. 



When, for instance, a red-blind person is confused in his perception 

 of those different sorts of light that to the normal eye appear as red 

 and green, we may conclude that he sees them both as one and the 

 same colour, but not what that colour is, as to its quality — whether it 

 is one of those just mentioned or a third — and whether, on the latter 

 supposition, that colour exists in the colour-system of normal-eyed 



