Bary centric Perspective. 183 



Analogous results maybe obtained for solid figures, substituting 

 the more general notion of homography for that of perspective, 

 as will more fully appear in the sequel. 



Remembering that M is the area of the primitive plane object, 

 it seems to result as an indirect inference from the preceding 

 theory, that whenever we can determine the area of an oval sec- 

 tion (whether the bounding curve be the whole or a part of the 

 curve of section) of an algebraical cone, then we can determine 

 the position of the centre of gravity of that oval in its own 

 plane by processes of differentiation only ; and, mutatis mutan- 

 dis, the same conclusion will admit of extension to solids bounded 



by algebraical surfaces ; so that 1 1 dx dy or I I I dx dy ds being 



given, subject to certain conditions of limit, I I (ax + by) dxdy, 



111 (a$ + by-\-cs)d9sdyds > subject to the Same conditions, become 



known by algebraical and differentiation processes only, and so 

 obviously for any number of variables*. 

 Cowley House, Oxford, 



July 1863. 



[To be continued.] 



so that 3M h?-\-2eah cos a,-\- a 2 e 2 — « 2 



— M fc+ea coses 



dk 



dc^_ ea sin cth _ x 

 dM. h+ea cos u, 



Ik 

 Avhere y and x are the coordinates of the point referred to in the text, if 

 we take the vanishing line and a line perpendicular thereto from the focus 

 for the axes of x and y. Consequently, if we remove the origin of coor- 

 dinates to the centre of the ellipse, preserving the directions of the axes, 

 and call x', y' the new coordinates, we shall have 



, . a 2 e 2 sin cc cos a, 



x ==aesmx—x= — - , 



h-\-ae cos x 



a(\ — e 2 (sinoc) 2 ) 



y' =h 4- ae cos cc-^ y = — j—. , 



y ' v h+ae cos a 



y' 1 — e 2 (sin ct) 2 



x' e 2 sin oc cos cc 

 which may easily be shown to he the equation to the diameter drawn to 

 the point of the ellipse where the tangent is parallel to the vanishing line ; 

 and consequently the perspective of the centre of gravity of the original lies 

 in this diameter, as evidently it ought to do, since every infinitesimal slice 

 of the qualiform area contained between parallels to the vanishing line is 

 of uniform density throughout, and is bisected by the diameter conjugate 

 to the direction of that line. 



* The inference made hesitatingly in the text, upon further reflection 



