Bary centric Perspective, 181 



primitive from the vanishing line becomes immediately known 

 by a process of differentiation when the area of the primitive is 

 expressed as a function of the distance of any arbitrarily fixed 

 point in the plane of projection from the vanishing line. For 

 if this area, which is the moment of the qualiform projection in 

 respect to the vanishing line, be called M } and the mass of the 

 same be termed Q, and if A, d be the distances of the origin and of 



M 



the centre of gravity from the vanishing line, we have d=-^- s 



H 

 where 



Vo (A-rsin0) 2 A' 



1< f 2 * tHO ( 1 1 1\ 



U ~3^J (h-r sin 6fh\h--r sin 6 + /i-rmn0~*' h) ; 



hence 



n- ldM 

 *" S"3P 



and 



dM 

 dh 



Thus, e.g., if we wish to find the perspective position of the centre 

 of gravity of the primitive of a given elliptic projection we have 

 found in a preceding foot-note, 



M= i a(A 2 + 2^ecosa + «V-« 2 )" i ; 

 hence 



«= r~; i 



n-t-ae cos a 



or, calling II the radius of the circle concentric with the given 

 projection, and having with it a common tangent parallel to the 

 vanishing line, and H the distance of the centre of this circle 



distances of the eye and P from the line of O, and d of the eye from the line 

 of P, then 



0:P::rfH:(H-£) 2 . 

 2. If be a plane element, P its perspective, H, h the distances of the 

 eye and P from the plane of O, and rfthe distance of the eye from the plane 

 of P, then 



0:P::dH 2 :(H-£) 3 . 



These formulae would become necessary in applying (as might be done 

 perhaps advantageously) in some cases the integral calculus to the quanti- 

 fication of curved lines and surfaces by a perspective method more general 

 than the one in ordinary use, which is essentially a method of orthogonal 

 projection. 



