Now let us suppose all the Lines in the diagram to extend to infinity both ways: and let us call the Infinite-Sector, lying between -produced and the upper part of the revolving-Line, 'No. 1'; the semi-Infinite-Strip (I mean, by 'semi-Infinite,' that it is terminated at one end), lying between the Lines -produced and -produced, and bounded at the right-hand end by , 'No. 2'; the surface, lying between the Line -produced and the lower part of the revolving-Line, and bounded at the left-hand end by , (which will be a Triangle, so long as -produced and the lower part of the revolving-Line continue to intersect, and will become a semi-Infinite-Strip when they cease to intersect,) 'No. 3'; and, with regard to the Infinite-Sector lying between the Line -produced and the lower part of the revolving-Line, let us call that portion of it, which lies above -produced (which portion will be a semi-Infinite-Strip, so [Pg 51]long as -produced and the lower part of the revolving-Line continue to intersect, and will become the whole Infinite-Sector if they should cease to intersect before the revolving-Line reaches the position ), 'No. 4'; and that portion of it, which lies below -produced (which portion will cease to exist as soon as these Lines cease to intersect) 'No. 5.' Let us now investigate the changes, in the areas of these 5 surfaces, caused by the changes in the position of the revolving-Line. First as to No. 1. This is clearly, throughout its history, an Infinite-Sector, whose vertical-angle is at first a right angle, and ultimately Zero. Also its area is at first one-quarter of the whole Infinite-Plane, and ultimately Zero. Also, so long as the upper part of the revolving-Line ranges between and , the area continues to be an Infinity of the second order; and, as the revolving-Line crosses the position , the area changes, from a very small (!) Infinity of the second order, to a very large Infinity of the first order. Similarly, as the revolving-Line crosses the position , the area changes, from a very small Infinity of the first order, to a very large Finite value. A little further on, it will of course become an Infinitesimal of the first order; and so on, through the other orders, till it finally reaches the value Zero. Next, as to No. 2. This is clearly, throughout its history, one-half of the Infinite-Strip lying between and , and is therefore an Infinity of the first order. Its area is a constant quantity, being unaffected by the revolving-Line.