168 Dr. Hirst on Equally Attracting Bodies. 



generally, if the same spiral were to move so that its plane con- 

 stantly touched, without sliding over the surface of any cone 

 whatever, having its vertex at the attracted point, it would gene- 

 rate a surface of the class under consideration. 



9. As a last conclusion, immediately deducible from equation 

 (1), we may cite the following : — 



The densities at corresponding points being the same, the result- 

 ant attraction upon the centre of a sphere of any quadrantal tri- 

 angle on its surface {or of any portion of such a triangle) is less 

 than that of the corresponding portion of any other surface what- 

 ever. 



The demonstration presents no difficulties. 



10. In order to arrive at the analytical expression of the first 

 theorem of art. 3, we propose next to determine, directly, the 

 requisite formulae in polar coordinates. 



Let ABB', A C C be two great circles of a sphere whose 

 planes are inclined to each other at the acute angle A. Further, 



let B C and B' C be two other arcs of great circles, whose planes 

 are at once perpendicular to each other and to ABB'. Then 

 in the right-angled spherical triangles ABC and AB'C we 

 have at once 



tan a = tan A . sin c, 



tan «'= tan A.sine', 



where a and c are the sides opposite to the angles A and C in 

 the triangle ABC, and a', c' the sides respectively opposite to 

 the angles A, C in A B' C But since the sum of, or difference 

 between c and c* is a right angle, we have, on squaring and adding 

 these equations, 



tan2«+tau2a' = tan2A (2) 



Again, if through B and B' we draw two arcs of great circles, 

 BI) = a and B'L'=a', perpendicular to ACC, we have, by well- 

 known formulae in spherical trigonometry, 



sin a = sin fl . sin C, 

 and 



cos A= cosa . sinC. 



Eliminating sin C from these equations, and performing the 



