180 Prof. Sylvester on the Principles of 



ing line of the quantities visibly represented in the barycentric 

 theory *. 



I have termed this a theory of barycentric perspective, because 

 it includes a method whereby the centre of gravity of a plane 

 figure is retained in perspective with the centre of gravity of its 

 projection ; by what has preceded, it appears that this may be 

 effected by regarding its projection, not as of uniform den- 

 sity, but of a density following the law of the inverse cube of 

 the distance. From this it follows that the distance of the per- 

 spective position in the picture of the centre of gravity of the 



circle, the area of the ellipse (regarded as made up of infinitesimal sectors 

 with the centre of the projection for their common vertex) becomes 



1. ,v3 



I 



imr 



h{h-rsin0f &8 /j_ /r.\2\J (#?_ r 2)1 



so that the area of any ellipse in a given plane, the perspective representa- 

 tion of which ellipse is a circle, will vary directly as the area of the circle, 

 and inversely as the cube of the tangent drawn to meet it from the ortho- 

 gonal projection of its centre on the vanishing line. More generally, if the 

 figure in the plane of projection be an ellipse with semiaxes a, b, eccentri- 

 city e, inclination of minor axis to vanishing line a,, and distance of one of 

 its foci from that line h, then calling V the area of the primitive and \i the 

 absolute ratio between a primitive element and its projection, we shall have 



.__ «(l-e 2 ) 



V =£l d6 7l ■ ,v, » where n- 



2h\ (h—rsmO) 2 1 + e sm 6 



This integration may be performed with extreme facility, and gives 



V= ! x7rab(h' i ±2heacosu-a 2 (\-e°)y%, 



say ~^ab, 



■where to find D we may use the following construction : — Draw a circle in 

 the plane of, and concentric with, the projection, and such that a common 

 tangent to the two shall be parallel to the vanishing line, and from the foot 

 of the perpendicular upon that line from the centre draw a tangent to the 

 circle, the length of the tangent so drawn will be D ; so that the area of any 

 ellipse will be to the area of its perspective projection as the product of the 

 square of the distance of the trace into that of the eye from the vanishing 

 line is to the cube of the tangent just described, — a very remarkable pro- 

 position in perspective, if new. By varying the origin of our polar coordi- 

 nates, as by taking it, for instance, at the centre of the projection or any 

 other point, we may obtain a new class of definite integrals of known values, 

 and which it might be exceedingly difficult to determine by any direct 

 method. It may be added that all ellipses in the same plane will bear 

 a constant ratio to their projections if those latter have a common tangent 

 parallel to the vanishing line, and their centres be in another line also 

 parallel to the same. 



* The above statements, combined with the varying law of frequency, 

 amount to the following propositions in perspective : — 



1 . If be a linear element, P its perspective representation, H, h the 



