3G6 PEOFESSOR A. R. FORSYTH ON THE DTEFERENTIAL INVARIANTS 
and the significance of some of them is known ; we proceed to consider this aspect of 
the invariants. 
In dealing with binariants, several methods are possible. There is the symbolical 
method. There is tlie method dependent upon the use of canonical forms for the 
various functions ; the complete expression of each binariant must be used through 
each operation ; in the present instance, the canonical form would arise by taking ^ 
and xp as the independent variables on the surface.* There is the method that 
depends upon the characteristic property of binariants, by which the leading term alone, 
being sufficient to determine tlie l)inariant uniquely, is used to replace the binariant. 
The last of these methods will be used. 
31. We denote by an arc of the curve (f) = 0, so that d/ds implies differentiation 
along the curve ; and we denote by d/dn differentiation in a direction on the surface 
perpendicular to the curve. Where no confusion will arise, we shall use x', x", 
m 
place of' 
dx d^x 
ds ’ ds^' 
. . . ; and so with quantities other than x. 
In constructing the fundamental quantities of order higher than the second, a 
normal section through the tangent to (p is drawn; successive derivatives of the 
curvature of this section at the point are constructed, and the values of the second 
derivatives of x and y are tliose connected with the geodesic property at the point.! 
Accordingly, it is effectively the geodesic tangent to (p that is drawn ; we shall denote 
by t an arc of this geodesic, so that d/dt inij)lies differentiation along the geodesic. 
As the curve and the geodesic touch one another, we have 
dll _ du 
ds ~ dt ’ 
when the quantities relate to tangential properties only; but 
du du 
ds dt 
is not zero when the quantities relate to contact of higher orders. Thus 
dx dx dy dy 
ds dt ’ ds ~ dt ’ 
but 
d / 1 
ds \ p 
d^l 1 
dt 
f / ’ 
1 . 
wliere , is the circular curvature of the geodesic tangent, is not zero 
p 0 0 5 
* This method is used by Daeboux, ‘ Theorie g^n^rale des Surfaces,’ vol. 3, p. 203. 
t See the paper quoted in § 4. 
