April 11, 1859.] HERSCHEL'S NEW PROJECTION OF THE SPHERE. 175 



180° on either side of it, and thus affords, what no other projection 

 does, an equally clear and perspicuous representation, not only of 

 the Indian, Atlantic, and Pacific Oceans, and the whole of the old 

 and new Continents and Australia, but also of the North Polar basin 

 — somewhat unduly enlarged, it is true, but exhibiting the whole 

 coast-line infinitely less disfigured than in the Mercator charts. 



The projection in question having been the direct result of a 

 general mathematical inquiry into the subject, suggested by the 

 consideration of Colonel James's recent projection, which takes in 

 more than a hemisphere, I subjoin the steps of the investigation 

 which led to it. Colonel James's projection takes in at the very 

 extreme theoretical limit only about 132"^ of amplitude from the 

 centre to the circumference of the. map, or five-sixths of the face 

 of the sphere; and even when restricted to 110° of amplitude, or 

 two-thirds of the sphere, though very elegant and pleasing in effect 

 as a picture, yet gives a very considerable amount of distortion of 

 shape at the borders. The sketch marked (A) includes an ampli- 

 tude of 160° of North Polar distance, or 97-lOOths of the whole 

 surface of the globe, and exhibits no distortion, and on the whole no 

 more variation in the scale in the representation of areas for areas, 

 than can very well be tolerated. 



Should you consider this a fitting communication for the Geo- 

 graphical Society, I will beg the favour of your laying it before them ; 

 and meanwhile permit me to remain, my dear Sir Koderick, 



Very faithfully yours, 



J. F. W. Herschel. 



In this paper the author investigates the general mathematical 

 expression for the co-ordinates of any point in the projection 

 in terms of the co-ordinates of the corresponding point on the 

 sphere. The condition, that any infinitesimal rectangle on the 

 sphere and its projection must be similar, leads to a differential 

 partial equation of the second order, the solution of which gives 

 rise to two arbitrary functions, entering into the expressions for 

 the co-ordinates of the projected point. 



These functions " being subject to no restriction, it is evident 

 that we may superadd to the general conditions of the problem any 

 which will suffice either to determine altogether or to limit the 

 generality of the arbitrary functions in the view of obtaining con- 

 venient forms of projected representation. Suppose, for instance, 

 we assume, as a condition, that the projected representations of all 

 circles about a fixed pole on the sphere shall be concentric circles 



