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298   Art of Travel.

certainly within the points of the compass, P s and P R. Draw the circumscribing parallelogram, G L H E m, whose sides are respectively parallel to P s and P R. Join L M. By the conditions of this problem, the path must somewhere cut the circle E D r ; and since L m cuts L H, which is a tangent to it, it is clear it must cut every path-such as a a, parallel to L H, or to P R-that cuts the circle. Similarly, the same line, L m, must cut every path parallel to P s, such as b b. Now, if L M cuts every path that is parallel to either of the extreme directions, P R or P s, it is obvious that it must also cut every path that is parallel to an intermediate direction, such as c c, but

P   PH   PD

L-cos HPL- Cos J RPS;

the consequence of which is that P L exceeds P D by onesixth, one-half as much again, or twice as much again, according as R P s = 60°, 90°, 120°, or 140.°

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.

 Extreme length of Road it may be necessary to travel. Probable length of Road it will be necessary to travel. Knows A, alone Knows A; and B to within 7 times the "least distance." 2 times the "least distance." 8 points. Not C. 44   11 2} Knows A; and B to within 12 points. Not C. 7¢   11 24 Knows A and C; and B to within 8 points .. 14 4 Knows A and C; and B to within 12 points.. 24   11 l4 Knows A and C; and B to within 13 points 3f   11 l4

* The words "least distance," mean the least distance that the traveller can specify with absolute confidence, as that within which the path, &c., he wishes to

regain, is situated.

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