182 Prof. Sylvester on the Principles of 



H 2 — R 2 

 from that line, d= ==- — , an equation the geometrical inter- 

 pretation whereof is readily obtained. 



More generally, if we take as cos a +y sin a-— h = as the equa- 

 tion to the vanishing line, using, as before, M to denote the 

 moment of the qualiform projection in respect to that line (well 

 worthy in this theory of being termed the principal moment), or, 

 which is the same thing, the area of the primitive, and take M* for 

 the moment of the same in respect to the axis of y } we shall have 



M=ff dxdy 



J J (#cos« + ysin«— hf 



M = ( Y dscdyx 



X J J(#cosa + ysina — h) 4 ' 

 from which it is easy to deduce 



M^cos^M + l^M^sin^M; 



M 

 and consequently -~ —h cos a, which is the distance of the per- 



spective of the centre of gravity of the primitive in the direction 



of x from the foot of the perpendicular from the assumed origin 



o ■ ™ • dU 

 ocosa.lvl -j-sma-y— 



upon the vanishing line, will be -rr? . And thus 



~dh 

 we are led to the remarkable proposition, that when we know 

 the area of the primitive in terms of the parameters of its vanish- 

 ing line, we can completely determine the perspective position 

 of its centre of gravity by means of processes of differentiation 

 only ; so that a method closely akin to (if not identical with) that 

 of potentials in the theory of attraction has a necessary place 

 also in the theory of perspective. 



If, as is most convenient, we fix the perspective of the centre 

 of gravity of the object figure by its distance from the vanishing 

 line and its distance from the line through the origin perpendi- 

 cular to the vanishing line, we see, by making a successively zero 



7T . 



and -^ in the above formula, that these distances are 



-M 

 ■j and — respectively*. 



dh dh 



* In the case of the ellipse, we have found in a preceding foot-note, 

 M=fx(h 2 +2aehcos*+a 2 e 2 -a 2 )%> 



