Map projections . ed. Whatever form may be given to k, the constant of the cone nis the reciprocal of k. For equating, as usual, the two expressions for the lengthof an element of parallel, we have 1\. 6 = R cos ,. AX, whence n = -r—«?-? (2). It was shown by Albers that if the general expression forthe radius of any parallel is r2= 2Jtf2/£(sin ^ - sin ty + r^ (3) and k = —— r- (4), sin —— cos ^ — 2 2 then the projection is equal-area. OF PROJECTIONS 9i To construct the projection we must first decide upon theparallels which we shall choose as standard ; then from theirlatitudes compute k, and
Map projections . ed. Whatever form may be given to k, the constant of the cone nis the reciprocal of k. For equating, as usual, the two expressions for the lengthof an element of parallel, we have 1\. 6 = R cos ,. AX, whence n = -r—«?-? (2). It was shown by Albers that if the general expression forthe radius of any parallel is r2= 2Jtf2/£(sin ^ - sin ty + r^ (3) and k = —— r- (4), sin —— cos ^ — 2 2 then the projection is equal-area. OF PROJECTIONS 9i To construct the projection we must first decide upon theparallels which we shall choose as standard ; then from theirlatitudes compute k, and thence rt or Draw one of thestandard parallels, divide it truly, and obtain the meridians asusual. The radii of other parallels may be computed from theequation (3), or from the corresponding equation r-= 2R-k{sm (j)2— sin (/>)+ r2° (5). But the computation is simplified if we combine (3) and (5)and obtain, after some reduction, r2 = £ (ri> + ri) + 2R- (1 -£sin <£) (6).. Fig. 16. Conical Equal Area with two Standard Parallels (Albers). The scale along a meridian is dr kR cos $ ?(7), Rd r which is obtained very simply when once r has been scale along a parallel is given by rdO nr RcosdX Rcos(f) kRcos .(8). 92 THE SIMPLE MA THEM, TICS And by multiplying together (7) and (8) we have iu. *->ie areascale at any point rdrdOR cos d(f>d\ which shows that the projection is an equal area projection. But it will be noticed that the equal area property does notdepend upon the form adopted for k. In fact we have not, upto the present, except in (6), made any use of the expressionfor k given in (4). We have shown that for any value of k theprojection is equal area, in the sense that any two equal areasupon the Earth will be represented by two equal areas uponthe map. It remains, however, to be seen whether the areascale corresponds to the linear scale upon which the standardparallels are represented. Consider
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