pn, qn, are given, the motion can be traced completely from these equations.
pn, qn, each moving solely under its own internal forces, and therefore in accordance with equations (i).
pn, qn lie within a range such that p t is between p i and pi +dpi qi „ „ qi „ qi +dq i, and so on.
A few of the large streams may, when in flood, spr.ead out in a temporary shallow sheet qn a dead level of clay, or playa, in a basin centre, but the sheet of water vanishes in the warm season and the stream shrinks far up its course, the absolutely barren clay floor of the playa, impassable when wet, becomes firm enough for crossing when dry.
qn differ from zero.
Their numerators are denoted by Pi, P2, their denominators by q,, q2, q3, We have the relations p n = an pn-1 +bn pn-2, qn = angn-1 +bngn-2.
The difference between the continued fraction and the nth convergent is less than, and greater than a n+2 These limits qn may be replaced by the following, which, though not so close, are simpler, viz.
and q (q + q +l qn +i).
qn-1 gnqn-1 qn qn-1' so that its value cannot oscillate.
n qn qn-i - () gngn-1, and the limit of the right hand side is not necessarily zero.
., an, b 21 ..., b n determined by the relations p n = a n pn-1 + b n pn-2, qn = a ngn -1 + bngn-2, with the conditions p1= at, q 2 = a2, q, = 1, 0) =0, have been studied under the name of continuants.
, p, ,, and for their denominators any assigned quantities ql, q2, q 2, The partial fraction b n /a n corresponding to the n th convergent can be found from the relations pn = anpn -I +bnpn -2 1 qn = anq,i l +bngn-2; and the first two partial quotients are given by b l =pi, a1 = ql, 1)102=1,2, a1a2 + b2= q2.
, qn are all unity, we find the series u 1 +u2+.
13 r qn ui, c It is generally possible so to arrange the method of observation as to eliminate the effect of an error in " the reading for coincidence of the webs " from the results.
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