1918]
ON THE THEORY OF LUBRICATION
529
If wo introduce the, value ot'/3, the equation of the. curve may be; written
h=L*'*..................................(2S)
When we determine k so as to make P a maximum, we get k~ 2'3, and
P = 0105 fJj(,r°\ ...........................(29)*
/'•ia
again with an advantage; which is but small.
Jn all the cases HO far considered the thickness h increases all the. way along the length, and the resultant pressure is proportional to the square of this length (c). In view of some suggestions which have been made, it is of interest to impure what is the effect of (say) r repetitions of the same curve, as, for instance, a succession of inclined lines AttCDKF (Fig. T). It
appears from (H) that It has the same value for the aggregate as for each member singly, and from (5) that the increment, of ]> in passing along the series in r times the increment due to one. member. Since the former increment is Hero, it follows that the pressure! is zero at the beginning and end of each member. The circumstances are thus precisely the mime for each member, and the total pressure is r times that, duo to the first, supposed to be isolated. Smt if we imagine the curve spread once over the entire length by merely ine.reasing the scale of a,1, we. see that the. resultant pressure, would be increased ?•" times, instead of merely r times, Accordingly a repetition of a curve m very unfavourable. But at this point it is well to recall that we are limiting ourselves to the, case of two dimensions. An extension in tho third dimension, which would suffice for a particular length, might bo inadequate) whcsn this length IB multiplied r times.
The forms of curve hitherto examined have been chosen with regard to practical or mathematical convenience, and it remains open to find the form which according to (5) makes P a maximum, subject; to tho conditions of a given length and a given minimum thickness (A,) of the layer of liquid. If wo suppose that h becomes h + Bh, where S is the symbol of tho calculus of variations, (8) gives
(so)
and from (7)
BP r
sss I
a//
.(31)
[• P iriere&MS rapidly from zero when k~l to the maximum given by (29), and then decreases slowly (P = Q'Ulfj.Ue*/hi*, when k*i). W. F. 8.]
R. vi. 34tor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRdŁ. For points in a plane