" (3.) flaying ascertained the projections, N, a, of one of the imints of the required common perpendicular, it will be sufficient to (Attain the poijection of the perpendicular itself, to draw through the 'stints IC, a, the right lines N t', tt p, perpendien:ar to the respective traces A E, F and the parts N e and a p of these perpendiculars, comprised between the projections of the two given right lines, will be the pro jections of the required shortest distance.
(1.) In conclusion, if the size of this shortest distance be desired to be known, it may be constructed by the process of Fiya•e 3.
"The consideration of a cylindric surface touched by a piano, was not essential to the solution of the preceding question. After having supposed a plane parallel to the two giveu right linen, we might through each of these lines have drawn to such plane, at perpendicular place ; and the inter section of these two planes would have been the direction of the required shortest distative. We content ourselves with announeing this second method, and advise the reader to seek its const•uctien by %•ay of exercise.
In the several questions which we have resolved relative to planes tangent to curved surfaces, we have always sup posed the point, through which the tangent plane should be drawn, to be taken un the sin-thee, and to be itself the point of contact : this condition alone sufficed to determine the position of the plane. But it is ditferent when the point through which the plane should pass is taken out of the surfitee.
" In order to determine the situation of a plane, it must satisfy three several conditions, each equivalent to that of passing through a given point. Now, in general, the pro perty of being tangent to a given curved s.urfitee, when the point of contact is not indicated, is only equivalent to one of these conditions : if therefore, we propose to determine the position of a plane by conditions of this nature, we shall generally have occasion for three. For instance : suppose three curved surfaces to be given, and that a plane be tan gent to one of in any point whatever; we can conceive that such plane would move around the surface, without ceasing to touch it : it would do so in every direction ; only the point of contact would shift its situation on the surface, in proportion as the tangent plane changed its position; and the direction of the point of contact would be similar to that of the motion of the plane. Suppose this movement to be
made in a certain direction till the plane meet the second surface, and touch it in a given point : then the plane would be tangent to the two first surf:tees at once, and its position would not yet be fixed. Indeed, the plane may be supposed to turn about the two surfaces, without ceasing to touch them both. It will no longer, however, be free to move in c% cry direction, as before, but will be confuted to one only. In proportion as the plane changes its position, the two points of ernitact will move each upon the surface to which it belongs ; so that it' a right line be conceived as passing through those points, their movements will be in the same direction with respect to such line, when the plane touches the two surfaces on the same side, and they will be in a contrary direction, when it touches one surface on one side, and the other on the contrary side. Lastly, imagine this motion, which is the only one that can now take place, to be continued till the plane touch the third surface in a certain point ; then its position will become fixed, and it can no longer more without ceasing to be tangent to one of the three surflices.
I fence we may perceive, that to determine the position of a plane by means of indeterminate contacts with given curved surfaces, we shall generally require three such star faces. Thus, were it proposed to draw a tangent line to a given curved surface, this condition would be equivalent to only one of the three to which the plane is callable of answering : we might, again, take two others, at Illeastu•e, and for example, make the plane pass through two given points, or, w hich is the same thing, along a given right line. Were it essential that the plane should be tangent to two surfaces at once, two conditions would be fulfilled ; there would remain but one to be disposed of and the plane could only be brought to pass through one given point,—Lastly, when the plane touches three given surfaces at once, there remains no longer any condition to be disposed of; its posi tion is determined.