Grinten — Projection of the Whole Earth? s Surface. 363 



wherein u denotes the radius of any meridian, v the distance 

 of its center from the central meridian and w the linear longi- 



m1 . \-io 2 \+w 2 .8v l+io 2 82V 



tude. There we have v= — - — ; u=— - — and _- = — — . -=-; 



2w ' 2io 8\ 2w 2 8\' 



8u 1—w 2 8w /?n\ 1 ( m \ 



— - = — . r— and substituting k — 1 — 1 , where 1 — \ 



8\ 2w 2 8\ * \wj cos <£' \ w ) 



becomes indefinite = — - at the central meridian, and find : 

 ~) = 1-<P; * 1=0 =^ or finally, 



©- 



(111) %_o = — • — — y - sec <£. 



_^ (1 +0) yT^— (1 -c) a/I+ c 



For the deformation A' at the marginal meridian we get by the 



above general formula: #, . on = — - — -.from which we 



a=180 cos <f> cos 



-. . t /t»x ' . /i 2c?— y(l + c? 2 ) c— y . 



derive: by (B) sm (9 = T _^ (2y _ J } = ^ the general ex- 



pression : (C) k ?=m = (1 _^ )2 . ^ =0 and for our y = — : 



(IY) A;, 1fin = — , sec <£ = — sec <f>. 



V / a=180 (2 _ c)a / 1+c ^ 5 ^ 



The deformation of area at the central and marginal merid- 

 ians is defined by 



S Z=0 = h=0 ' h=6 and S a = 180 = ^=180 • ^=180 • cos 0> 

 respectively. 



The maximal and minimal linear alterations are then repre- 

 sented by the conjugate diameters a and b of an infinitely 

 small ellipse — called the indicatrix — which is produced by an 

 orthogonal projection of an infinitely small circle — circum- 

 scribing a point ( — ) — from any curved surface upon a plane. 

 These diameters are defined by the relations : 



a . b — s—h . k . cos 6 

 and the maximal distortion of angle (2o>) by 



- a ~^ /i /h 2 + k 2 — 2hkco&~0 

 Sin W ~~ a + b " * h 2 + k 2 + 2hk cos ' 



The y = cl projection coincides with ours for d = ^- ; 



6 = 18° ; cf> = 80°, 46 (sectio divina). 



