289 
represents a curve of the p” order, considered as the locus of the variable 
point P; and that any homogeneous equation of the g dimension, of the 
form 
F, (1, m,n) =0, 
may in like manner be considered as the tangential equation of a curve of 
the q™ class, which is the envelope of the variable line LMN. But any 
examples of such applications must be reserved for a future communi- 
cation. Meantime, I may just mention that I have been, for some time 
back, in possession of an analogous method for treating Points, Lines, 
Planes, Curves, and Surfaces in Space, by a system of Anharmonic 
Co-ordinates. 
8. As regards the advantages of the Method which has been thus 
briefly sketched, the first may be said to be its geometrical interpreta- 
bility, Im a manner wnaffected by perspective. The relations, whether 
between variables or between constants, which enter into the formula of 
this method, are all projective ; because they all depend upon, and are 
referred to, anharmonic functions, of groups or of pencils. 
9. In the second place, we may remark that the great principle of 
geometrical Duality is recognised trom the very outset. Confining our- 
selves, for the moment (as in the foregoing articles), to figures in a 
given plane, we have seen that the anharmonic co-ordinates of a point, 
and those of a right line, are deduced by processes absolutely similar, the 
one from a system of four given points, and the other from a system of 
Sour given right lines. And the fundamental equation (Ix + my +nz=0) 
which has been found to connect these two systems of co-ordinates, is 
evidently one of the most perfect symmetry, as regards points and lines. 
An analogous symmetry will show itself afterwards, in relation to points 
and planes. 
10. The third advantage of the anharmonic method may be stated to 
consist in its possessing an ¢ncreased number of disposable constants. Thus, 
within the plane, trilinear co-ordinates give us only siz such constants, 
_ corresponding to the three disposable positions of the stdes of that assumed 
triangle, to the perpendicular distances from which the co-ordinates are 
supposed to be proportional ; but anharmonies, by admitting an arbitrary 
unit-point, enable us to treat two other constants as disposable, the num- 
ber of such constants being thus raised from s7x to eight. Again, in space, 
whereas guadriplanar co-ordinates, considered as the ratios of the dis- 
tances from four assumed planes, allow of only éwelve disposable constants, 
corresponding to the possible selection of the four planes of reference, 
anharmonic co-ordinates, on the contrary, which admit either five planes 
or five points as data, and which might, therefore, be called guinguipla- 
nar or guinguipunctual, permit us to dispose of no fewer than fifteen con- 
stants as arbitrary, in the general treatment of surfaces. 
(To be continued.) 
