for Maps applying to a large extent of the Earth's Surface. 307 



projection $" by G. B. Airy, Esq., Astronomer Royal, which 

 appeared in the Philosophical Magazine for December 1861, 

 and in examining the numerical results given in the Tables in 

 which the relative advantages of the Projection by Balance of 

 Errors, by Equal Radial Degrees, by Unchanged Areas, and in 

 the Stereographic (attributed to Hipparchus), and my Projection 

 of two-thirds of the sphere, I was struck with the fact that the 

 numbers given as representing the Radial distances from the 

 centre of the Map, the Exaggeration of the projected areas, and 

 the Distortion of the form, did not show such advantages in 

 favour of the Projection by Balance of Errors as I was naturally 

 led to expect from the ingenious method employed by Mr. Airy 

 for obtaining them. 



I therefore requested Captain Alexander R. Clarke, R.E., to go 

 through the mathematical process employed by Mr. Airy, and 

 examine the numerical results given in the Tables. I subjoin 

 the result of Captain Clarke's examination ; and it will be observed 

 that, from a mistake inadvertently made in one of the constants,, 

 the projection by Balance of Errors has greater advantages than 

 Mr. Airy has given it in his Tables. 



The fundamental equation of this very beautiful method of 

 development is readily obtained in the following manner. Let 

 P be the point on the sphere which is to be the centre of the 

 map, and let Q be any other point on the sphere such that the 

 arc PQ = ; if Q' be the representation of Q on the Development, 

 PQ' = r. Suppose a very small circle, radius a>, described on 

 the sphere having its centre at Q, then the representation of 

 this circle in the map will be an ellipse having it's minor axis 

 in the line PQ' and its centre at Q'. The lengths of the semi- 

 axes will be 



dr (o 



w dd' r \ 



The differences between these quantities and that (©) which 

 they represent are 



CO 



(J- 1 )- "Gib- 1 ) 5 



and the sum of the squares of these errors is the measure of the 

 misrepresentation at Q'. The sum for the whole surface from 

 = to 6= ft is proportional to 



f{(l-'H^-0>». 



which is to be a minimum. 



