Page:The New International Encyclopædia 1st ed. v. 17.djvu/471

SAILINGS. 431 SAILINGS. gain in latitude and in departure when the ship's track is made up of several pieces, the whole track being called a 'traverse.' In Figure 2 V is the i)oint of departure and H the point of arrival; and W.MiFfiH is the ship's track. The total gain in latitude is equal to (^1 — k+k — '.+^). The total gain in dc- a certain number of miles measured along the parallel of latitude, then p is equal to cosL minutes of longitude, or if we call the dilTerenee of longituile 1). we have D = ;)secL. Having obtained the value of p by means of the lormulu! parture is equal to (Pi+Pi + Ps — P,+p^). Each value of p and I may be computed from its own triangle. In sailing due east or west along a parallel of latitude the difference of latitude (i.e. I) is zero and p =: d =^ distance sailed. But p is expressed in nautical miles. To. determine how many minutes of longitude to which it corre- sponds, we must determine the length of a minute of longitude. 0/ B Fib. 3. In Fig. 3, W N E S is the meridian of the earth passing through the point P. OE = R = the equatorial radius of the earth. TP = r =: the radius of the circle of latitude passing through the point P. Circumference of circle of latitude 2rr Circumference at equator 2irR' Each of the circumferences is divided into the same number of minutes of longitude, therefore x' length of a minute of longitude at P r X ^length of a minute of longitude at equator R Since the earth is very nearly a sphere, we may without serious error assume it to be so. (See L.TiTrDE AND LoNoiTUnE. ) Then we have angle TPO = angle POE = L = latitude of P (near- r ly) ; also OP = OE (nearly) ; and cosL = ^ or x' = iccosL. If p ( = departure) correspond to of plane and traverse sailing, we find D by the formula D = psecL. The value of I, p. and D may be picked out of a table of right triangles such as is given in Bowditch's Saviijulur and other works of the kind, or the triangle may be solved in the usual trigonometrical manner. PAR.iiEL Sailing is a sjxcial case of plane sailing or traverse sailing in «hirh tlic course is east or west along a parallel of latitude. The formuUe may be deduced from those for traverse or plane sailing b}- putting C = HO". The latitude (L) used in the fiuegoing formu- lie is that of the point of departure. If the distance sailed is considerable and the change in latitude more than a few miles, it is evident that the resulting difference of longitude will be con- siderably in error, for the length of a minute of latitude at the latitude L ditTcrs from the length of a minute at L' (the latitude at the point of arrival ) . The exact average length of a minute of longitude is slightly greater than the mean of its lengths at the latitude of L and L' and slightly less than its length at the latitude of L + L' — ^ — . but the error is not large for ordinary cases, and it is customary to use the formula D = psec ( — s — )'• an<l this, together with I = <fcosC and p = rfsinC, which have already been given, constitute the formula" used in eom- ptiting a ship's position by "dead reckoning' (q.v.) when the latitude and longitude of the point of departure and the courses and distances sailed are known. Thus, suppose a ship leaves a place of which the latitude is .30° N. and the longitude 00° W. and sails northeast 100 miles and then S.S.E. 00 miles; required, the latitude and longitude of the place of arrival. The fol- lowing table is prepared: COrBSE Distance (d) DIB. lat. (1) Dep. (P) nitr. Ions. (D) X.E S.S.E 100 60 ■1-T0.7 —65.4 —70.7 -28.0 — «.! — 3«.8 -1-16.3 —93.7 -108.9 The latitude of the place of arrival is there- fore 30° 15' 18" (30° + 15'.3), and the longitude