539 
equation (21) expresses the condition necessary, in order that a variable 
plane [1..s8| may pass through a gwen point («..v); and in this view, 
the formula (21) may be considered to be the Tangential Equation of that 
gwen Point. Thus the very simple equation, 
VER) (45) 
expresses the condition requisite for the plane [7..s] passing through 
the given point (10000), or a (27); and it is, in that sense, the tangential 
equation of that point: while m = 0 is, in like manner, the equation of 
B, &c. This being understood, if we suppose that Fr and ¥’ denote two 
given, linear, and homogeneous functions of the coordinates /..s of a 
variable plane II, we may consider the four equations, 
r=0, FV=F, W=0, WY =hr, (46) 
as the tangential equations of fowr collinear points, Py, Py, Poy Ps, 
whereof the three first are entirely given, but the fourth varies with the 
value of the coefficient 4, although always remaining on the line A of 
the other three; and then it is easy to deduce, from the formula (81), 
by reasonings analogous to those employed in [ 11. ], the following anhar- 
monie of the group: 
(P P,P)Ps) =k = = (47) 
We have therefore this new Theorem, analogous to one lately stated : — 
“The Quotient of any two given, homogeneous, and linear Functions, 
of the Quinary Coordinates of a variable Plane, may always be expressed 
as the Anharmonie of a Group of Points ; whereof three are given and col- 
linear, while the fourth ts the Intersection of the variable Plane with the 
given Line on which the other three are situated.” 
[14.] For example, if we wish in this way to enterpret the quotient 
m:n, of these two coordinates of a variable plane Il, or | lmnrs | (28), as 
denoting the anharmome of a group of points, the three first points Po, Py, Pz 
of that group (47) have here for their tangential equations, 
n=0, m—n=0, m=0, (48) 
whereof the third has recently been seen [18.] to represent the given 
point 8, and the first represents in like manner another given point, 
namely c, of the initial system: while the second represents the point 
(0, 1, — 1, 0, 0), or briefly (01100), if, to save commas, we write 1 
for—1. To construct this last point, let us write 
a’ = (01100) = (10011), anda” = (01100); (49) 
then, by (18), these two new points a’ and a” are each collinear with 
B, ¢, or are on the /ine sc; and they are, with respect to that line (or to 
its extreme points) harmonically conjugate to each other, because the for- 
mula (31) gives easily, by the first symbol for a’, the harmonic equation, 
' (BA ca") = -—'1; : 7 (50) 
