2 ' REPORT—1863. 
result derived by Cauchy’s symbol just mentioned the coefficient of = in the ex- 
pansion of the given function in descending powers of (x). Now let 
F (2, Y; = = 45) denote any function of 2, y, and its differential coeffi- 
cients. This is sometimes written F(z, y, ¥:, Ya++++ Yn). Now there are in- 
vestigations in which we require the value of a where yn is put equal to zero 
na 
after the differentiation is performed. The writer has found a symbol such as 
Z™ to denote this, of great utility in the treatment of differential equations; and it 
will be observed that it belongs to the second class of symbols here mentioned. 
On the Quantity and Centre of Gravity of Figures given in Perspective, or 
Homography. By Professor Sytvester, /.2.S, 
In the first instance, the author showed how to find the point in the perspective 
representation of a plane figure into which the centre of gravity of such figure is 
projected. For this purpose it is only necessary to be furnished with the direction 
of the vanishing-line corresponding to the plane of the object put into perspective. 
The rule for finding the point in question is the following: every element of the 
picture is to be charged with a density equal to the inverse fourth power of its 
distance from the vanishing-line; the centre of gravity of the figure so charged 
will be the point required, and may of course be found by the rules of the integral 
calculus. 
Next, as {o the area of the unknown object. To determine this another datum 
(but only one other) is required besides the direction of the vanishing-line, which 
may be termed the constant of perspective, being determined when the position of 
the eye and that of the object-plane in reference to the picture are given. This 
constant is the product of the eye’s distance from the vanishing-line into the square 
of the distance of the intersections of the object- and picture-planes from the same 
line. If now every element of the picture be charged with a density equal to the 
constant of perspective divided by the cube of the element’s distance from the 
vanishing-line, the mass of the figures so charged will be the area of the unknown 
object-figure. 
The author then proceeded to show how the area and the perspective centre, by 
aid of the preceding principles, admit of being reduced to depend on one single 
integral, closely analogous to the potential used in the theory of attractions to which 
he gives the name of polar potential. The polar potential of a plane figure in 
respect to a given line is defined to be the sum of the quotients of the elements by 
their respective distances from the line, and consequently the polar potential of the 
picture in respect to a vanishing-line in its plane becomes a function of the two 
parameters by which its position may be determined. The parameters which the 
author finds most convenient to employ are the distance of the yanishing-line from 
an arbitrary fixed point in the picture and the angle which it makes with a fixed 
line therein. 
The author then supplied the formule (which are of a very simple character) for 
calculating the area of the object and the coordinates of its perspective centre of 
gravity, by means of differentiation processes performed upon the polar potential of 
the picture treated as a function of these parameters. He afterwards proceeded to 
extend the same method to figures, plane or solid, connected by the more general 
relation known under the name of homography, of which the relation between 
figures generated through the medium of perspective is only a particular kind. In 
the case of a solid figure, its polar potential in respect to a variable plane becomes 
a function of three parameters ; and by means of differentiations performed upon it 
in respect to these parameters, the content and the coordinates of the point cor- 
responding homographically to the centre of gravity of a solid figure may be ex- 
pressed when its homograph and the position of a plane corresponding to the 
