MATHEMATICS: E. J. WILCZYNSKI 
251 
tk. The line PkPk+i will intersect 4 and tk in two points, Ik and Tk, 
and the cross-ratio {I'k, Tk, Pk, Pk^i) turns out to be equal to 
3 
except for terms of higher than the first order in bxk. 
By a perspective correspondence the three points T'n-i, Pn~i 
of Pn-i B maybe projected into the points T'n-2, Pn-i of Pn-2 Pn-i- 
Let Bn-i be the point of Pn-2 Pn-i which, in this perspective, corresponds 
to B. Then project similarly /'n-2, T^n-2. Pn-2, Bn-i into the four 
points /'n-3. T'n-3, Pn-2, Bn-2 of Pn-3 , and contlnuc in this way. 
We shall finally obtain upon the line APi sl point Bi determined from B 
by this sequence of perspectives. As n grows beyond bound, Bi will 
approach a limiting position Q on the initial tangent of the arc AB. 
The cross-ratio 
k = (/„, To, A, Q) (7) 
will be the limit which the product 
approaches when n grows beyond bound. Consequently we find 
log k = f </e,Xx) dx. (9) 
This equation contains the desired interpretation of the integral p. 
From a theoretical point of view the expression (5) for the integral p 
is the simplest and most general. We shall however give, in conclusion, 
three other expressions for p in terms of more familiar variables. 
If the curve is given by means of its cartesian equation in the form 
3^ = / we may write 
P = 
I — 7 - — dx, (10) 
where y' = dy/dx, y" = d^y/dx,^ and so on. 
If the curve is given by means of parametric equations of the form 
X = (p{s), y = \p(s), where s denotes the length of arc, and if r is the 
radius of curvature at the point which corresponds to the value s of 
the parameter, we find 
p-^\' ^f^^^^+4r:-ds, (11) 
where / = dr/ds, r" = d^r/ds^, etc.. 
