530 ON THE THEORY OF LUBRICATION [428 the integrations being always over the length. Eliminating 8H, we get /Q0, x ............. (32) The evanescence of £P for all possible variations 8h would demand that over the whole range either $lrsxdx j „-„. XOQ\ "- TJFvJT or h"^H' ..................... (33) But this is not the requirement postulated. It suffices that the coefficient of Bli on the righfc of (32) vanish over that part of the range where h>hl, and that it be negative when h=-hif so that a positive Bh in this region involves a decrease in P, a negative 8/1 here being excluded a priori. These conditions may be satisfied if we make h = hl from x = 0 at the edge where the layer is thin to x = cl, where Cj is finite, and k = %H over the remainder of the range from cx to GI + c2, where GI + cz = c, the whole length concerned (Fig. 2). For the moment we regard Cj. and c2 as prescribed. In, 0 Fig. 2. For the first condition we have by (8) ,, i -r-r C-l/fil T" so that 0^ = ^(2^-3), ........................... (34) determining k, where as before k = h2/Jil~ The fulfilment of (34) secures that h = §H over that part of the range where h = h2. When h = hl} h — $H is negative ; and the second condition requires that over the range from 0 to Ci be positive, or since c^ is the greatest value of as involved, that / A-8 xdx ~ cj h~3 dx = + ...................... (35) The integrals can be written down at once, and the condition becomes J<*<c/(c12, ........................... , ..... (36) whence on substitution of the value of GS/CJ from (34), &(2&-3)2>l ............................... (37)* If k be such as to satisfy (37) and Ca/Cj be then chosen in accordance with (34) and regarded as fixed, every admissible variation of h diminishes P. But the ratio c.2/Ci is still at disposal within certain limits, while c-i + C2 (= c) is prescribed. [* This inequality may be written (A; -1) {4 (fc-l)2-3}>0; showing that, since /c>l and the conditions are satisfied when (37) becomes an equality, w'e must have k -f: 1 + -5- -j: 1 "866. W. F. S.]lowly (P = Q'Ulfj.Ue*/hi*, when k*i). W. F. 8.]