Report of the Board of Regents Volume 1863 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1864 Excerpt: ...all these sums will have the same value. Consequently, if we add them together, and let n denote the number of positions of the system of the two planes, the total sum will be equal to n times the value of one of the partial sums, or to n 1 V Now, this total sum is that of all the curvatures V C '/--, --, --, --, &c., in number 2n, corresponding to all the sections determined P P' P" P by the two planes. If, then, we divide the above equivalent quantity by 2-, the result--I 1-. will represent the mean of all these curvatures. Now, 2 V P P J as this result is independent of the value of -, or of the number of positions occupied by the system of the two planes, it will be coually true if we suppose / this number to be infinitely great, or, in other words, if the successive positions of the system of the two planes are infinitely approximated, and consequently if this same system turns around the normal in such a manner as to determine all the curvatures which belong to the surface around the point in question. The quantity--I--I represents, then, the mean of all the curvatures of the surface at the same point, or the mean curvature at this point. Now if, in passing from one point of the surface to another, the quantity--H retains the same value, i. e., if for the whole surface we have--= C, this sur P P face is such that its mean curvature is constant. Considered hi this purely mathematical point of view, the equation (4) has formed the object of the researches of several geometricians, and we shall profit by these researches in the subsequent parts of this memoir. Thus our liquid surfaces should satisfy this condition, that the mean curve must be the same everywhere..We can understand that if this occurs, the mean effect of the curvatures at ...