ME. M. W. CROFTON ON THE THEORY OF LOCAL PROBABILITY. 
199 
It is well to notice, with regard to the applications to integration of the theory laid 
down in this Paper, that the theorems thereby deduced in no way depend for their truth 
upon the doctrine of Probability, although it has been the occasion which has led to 
them. The apparatus of a system of equidistant parallels, revolving through constant 
angular displacements, which has been used in establishing their truth, is a strictly 
geometrical conception, and which, as here employed, may be viewed as a method in 
the Integral Calculus. A simpler species of reticulation, consisting of two systems of 
parallels, crossing at a finite angle, has already been used by Eisenstein and others in 
the Theory of Numbers and in Elliptic Functions. 
It will be borne in mind also that this apparatus of lines is used only as a correct and 
convenient representation of an infinite system of random lines, for the purposes of cal- 
culation. Of course it is not asserted that all those random lines which are parallel 
to a given direction will be equidistant, or* that there will be none of the random lines 
intermediate in direction between 0 and 0 + ^3. Just as an infinity of points arranged 
in horizontal rows and vertical columns will faithfully represent, for the purposes of 
calculation, an infinity of random points, so will the above apparatus represent the 
lines. Other arrangements, in either case, may easily be conceived which will represent 
them equally correctly, and which possibly will be found, in certain cases, more con- 
venient. Thus if an infinite plane be covered with points arranged symmetrically, the 
system of lines obtained by joining each pair of points will, undoubtedly, truly represent 
a system of random lines. 
It is unnecessary to point out, that if we can succeed in the difficult inquiry involved 
in extending the above methods to space, not only will the theory of probability be 
advanced, but various remarkable results in the Integral Calculus may be expected. 
