354 
p here denoting the variable point A-II, in which the fixed line A inter- 
sects the variable plane I ={ dmnr'. , 
20. All problems respecting intersections of lines with planes, &c., are 
resolved, with the help of the Fundamental Theorem (16) respecting the 
relation which exists between the anharmonic co-ordinates of point and 
plane, as easily by the present method, as by the known method of gua- 
driplanar co-ordinates (10) ; and indeed, by‘the very same mechanism, of 
which it is therefore unnecessary here to speak. 
But it may be proper to say a few words respecting the application 
of the anharmonic method to Surfaces (7); although here again the 
known mechanism of calculation may in great part be preserved un- 
changed, and only the interpretations need be new. 
21. In general, it is easy to see (comp. 7) that, in the present 
method, as in older ones, the order of a curved surface is denoted by 
the degree of its local equation, fixyzw) =0; and that the class of the 
same surface is expressed, in like manner, by the degree of its tangential 
equation, F(lmnr) = 0: because the former degree (or dimension) deter- 
mines the number of points (distinct or coincident, and real or imagi- 
nary), in which the surface, considered as a locus, is intersected by an 
arbitrary right line; while the latter degree determines the number of 
planes which can be drawn through an arbitrary right line, so as to touch 
the same surface, considered as an envelope. It may be added, that I 
find the partial derivatives of each of these two functions, f and F, to be 
proportional to the co-ordinates which enter as variables into the other ; 
thus we may write 
[D.f, Dif, Df, Duf ); 
as the symbol (15) of the tangent plane to the locus f, at the point (xyzw); 
and 
(DF, Dik, D,F, D,F), 
as a symbol for the point of contact of the envelope F, with the plane 
[Zmnr |: whence it is easy to conceive how problems respecting the polar 
reciprocals of surfaces are to be treated. 
22. As a very simple example, the surface of the second order which 
passes through the nine points, above called ancpEA’A,C’c,, is easily found 
to have for its local equation, 0 = f = xz — yw; whence the co-ordinates 
of its tangent plane are, /= D,f= 3, m=D,f=-—w,n=D,f=z, 
r= D,f =— y, and its tangential equation is, therefore, 0 = = In - mr, 
so that it is also a surface of the second class. In fact it is the hyperbo- 
loid on which the gauche quadrilateral ancy is superscribed, and which 
passes also through the point x; and the known double generation, and 
anharmonic properties, of this surface, may easily be deduced from either 
of the foregoing forms of its anharmonic equation, whereof the first may 
(by 18, 15) be expressed as an equality between the anharmonic func- 
tions of two pencils of planes, in either of the two following ways :— 
(Bc. AEDP) = (DA. BECP); (AB. CEDP) = (CD. BEAP). 
Erratum.—In line 5 of Art. 12, for a read wu. 
ee a ae 
