532 ON THE THEORY OF LUBRICATION [428 corresponding ratio F/P are both rather more advantageous in the arrangement now under discussion than for the simply inclined line. But the choice would doubtless depend upon other considerations. The particular case treated above is that which makes P a maximum. We might inquire as to the form of the curve for which F/P is a minimum, for a given length and closest approach to the axis of x. In the expression corresponding with (32), instead of a product of two linear factors*, the coefficient of Bh will involve a quadratic factor of the form JBxh + Ch2 + Dx + M + F, .....................(46) so that the curve is again hyperbolic in the general sense. But its precise determination would be troublesome and probably only to be effected by trial and error. It is unlikely that any great reduction in the value of F/P would ensue. Fig. 3 is a sketch of a suggested arrangement for a footstep. The white parts are portions of an original plane surface. The four black radii represent grooves for the easy passage of lubricant. The shaded parts are slight depressions of uniform depth, such as might be obtained by etching with acid. It is understood that the opposed surface is plane throughout. Fig. 3. [* This statement appears to be due to an oversight; We have in fact d (FjP) P2/(6Ai2 Z72) J/r-s dx = J Jr* ( - 27i + 3J5T) 6h { - 3 J ft-« dx J li^-x dx + 4 / Jr1 dx J h~*x dx + W»/(3/tJ7) J h~3dx - asFlfaU) J Jr^dx} dx, the integrations being over the length (c). Hence for a minimum of F/P the boundary may be taken as h = hi from x = 0 to x=cl, as 7i=7is = 3.H/2 from Cj + cg to c, and from ci to Ci + cg as an oblique line with an equation which must be made to coincide with the second factor equated to zero. This line must be continuous with the first at (clt h^, in order that over the latter the second factor may be positive, and it is inclined to the axis of x at an angle tan-x3.F/P. If there is a discontinuity 7t3 - h^ at ^=c1 + c2) and h3=k'h1, \ = lliz, where &>1, i<l, the condition 3-ff=27i3 yields by (8) The remaining two conditions, to be derived from proportionating the second factor to - \ + ?wc1 + h - mx, where mc2 = 1^ (1 - 1)11, provide two equations of the second and third degrees respectively in Cj/Cg, and lead to very complicated expressions. Without, however, including the oblique line, it may be shown that the two lines h=hi from x=Q to clt and h — hz=lchi ^°^ GI to Cj + Cg, with 2/i2=3JT, as on pp. 530, 531, make F/P a minimum when h^h^, "provided 4/c3-8fc2 + fc-3<};0, leading to fc <j: 2-06 approximately: since these lines have been shown on pp. 530, 531 to make P a maximum, they therefore also make F a minimum when k Jf. 2-06. With 7c=2-06, (44) and (39) make jyP=4-0137i3/c, P=-19469Mtfc2/7ia2. But, with F/P positive, the minimum value of (44) occurs at 7c = 2, when P/P=47i1/c, P=0-2/*D'c2//i12. Accordingly, as this value of F/P is not a minimum for all variations outside the straight line h=h1, the actual minimum value of F/P must be less than 47^/c. W. P. S.]ng there is some light inxxix. p. 128, 1874). I do not know the date of Thoulet's use of the solution, but suspect that it was subsequent to Sonstadt's.