1864] Prof. H. T. S. Smith on Quadratic Forms, 199 



result to those cases in which the ground whereon we stand is the plane of 

 the Horopter, it follows that, looking straight forward to the horizon we 

 can distinguish the inequalities and the distances of different parts of the 

 ground better than other objects of the same kind aud distance. 



This is actually true. We can observe it very conspicuously when we 

 look to a plain and open country with very distant hills, at first in the 

 natural position, and afterwards with the head inclined or inverted, looking 

 under the arm or between our legs, as painters sometimes do in order 

 to distinguish the colours of the landscape better. Comparing the aspect 

 of the distant parts of the ground, you nrill find that we perceive very 

 well that they are level and stretched out into a great distance in the 

 natural position of your head, but that they seem to ascend to the horizon 

 and to be much shorter and narrower when we look at them with the head 

 inverted : we get the same appearance also when our head remains in its 

 natural position, and we look to the distant objects through two rectangular 

 prisms, the hypothenuses of which are fastened on a horizontal piece of 

 wood, and which show inverted images of the objects. But when we invert 

 our head, and invert at the same time also the landscape by the prisms, we 

 have again the natural view and the accurate perception of distances as in 

 the natural position of our head, because then the apparent situation of 

 the ground is again the plane of the Horopter of our eyes. 



The alteration of colour in the distant parts of a landscape when viewed 

 with inverted head, or in an inverted optical image, can be explained, I 

 think, by the defective perception of distance. The alterations of the 

 colour of really distant objects produced by the opacity of the air, are well 

 known to us, and appear as a natural sign of distance ; but if the same 

 alterations are found on objects apparently less distant, the alteration of 

 colour appears unusual, and is more easily perceived. 



It is evident that this very accurate perception of the form and the 

 distances of the ground, even when viewed indirectly, is a great advantage, 

 because by means of this arrangement of our eyes we are able to look at 

 distant objects, without turning our eyes to the ground, when we walk. 



April 21, 1864. 

 Major-General SABINE, President, in the Chair. 



The following communications were read : — 



I. '^^On the Orders and Genera of Quadratic Forms containing 

 more than three Indeterminates.^^ By H. T. Stephen Smith, 

 M. A., F.R.S., Savilian Professor of Geometry in the University 

 of Oxford. Received March 22, 1864. 



Let us represent by a homogeneous form or quantic of any order 

 containiug n indeterminates ; by (a^^)), a square matrix of order n ; by 



