Pl = Ph/cos HPL = PD/cos 1/2 RPS;

The consequence of which is that P L exceeds P D by one-sixth, one-half as much again, or twice as much again, according as R P S = 60 degrees, 90 degrees degrees, or 140 degrees.

The traveller who can only answer the questions A and B, but not C, must be prepared to travel from P to L, and back again through P to M, a distance equal to 3 P L. If, however, he can answer the question C, he knows at once whether to travel towards L or towards M, and he has no return journey to fear.  At the worst, he has simply to travel the distance P L.

The probable distance, as distinguished from the utmost possible distance that a man may have to travel in the three cases, can be calculated mathematically.  It would be out of place here to give the working of the little problem, but I append the rough numerical results in a table.

[Table as referred to above].

The epitome of the whole is this:—­1.  If you can only answer the question A, you must seek for the lost path by the tedious circle plan; or, what is the same, and a more manageable way of setting to work, by travelling in an octagon, each side of which must be equal to four-fifths of P D.  (See fig. 2.)

[Fig. 2].

That is to say, look at your compass and start in any direction you please; we will say to the south, as represented in the drawing.  Travel for a distance, P D; then supposing you have not crossed the path, turn at right angles, and start afresh—­we will suppose your present direction to be west—­travel for a distance 4/10 of P D, which will take you to 1; then turn to the N.W. and travel for a distance 8/10 of P D, which will take you to 2; then to the N. for a similar distance, which will take you to 3; and so on, till the octagon has been completed.  If you know B to eight points, and not C, adopt the L M system; also, if you know A and C, and B to within thirteen points (out of the sixteen that form the semicircle), you may still adopt the L M system; but not otherwise.  A rough diagram scratched on the ground with a stick would suffice to recall the above remarks to a traveller’s recollection.

CACHES AND DEPOTS.

Caches.—­It is easy enough to choose a spot, which you yourself shall again recognise, for digging a hole, where stores of all kinds may be buried against your return:  neither is it difficult to choose one, so that you may indicate its position to others, or else leave it to a party who are travelling in concert, to find it out for themselves.  But excessive caution in the mode of depositing the stores is, in every case, required, as hungry and thieving natives keep watch on all the movements of a party; they follow their tracks and hunt over their old camping-places, in search of anything there may be to pick up.  And hyenas, wolves, wild dogs, and all kinds of prowling animals, guided by their sharp scent, will soon scratch up any provisions that are buried carelessly, or in such a way as to taint the earth.