APPLIED MATHEMATICS 187 



he uses w and wjg in one lecture, and m and mg in an imme- 

 diately following lecture — the former being in an elementary- 

 course, the latter in an honours course. 



In a letter to Nature, 109, 1922, 645-7, A. Gray makes 

 some interesting remarks on some well-known elementary 

 propositions in attractions, showing how immediate proofs 

 can be obtained, rigid and complete, but yet almost intuitive. 

 The proof by means of Gauss' theorem that a uniform spherical 

 shell attracts at any outside point like an equal point mass 

 at the centre, is of course the common property of all teachers 

 and most students of the attraction theory. To prove the same 

 for the potential. Gray uses elementary cones through the 

 point where the shell is cut by the radius to the outside point 

 considered, considering the elements of area cut off on the 

 concentric sphere through this point. It is also shown how 

 to apply the method to ellipsoidal shells. Another problem 

 taken is that of the force between the two hemispherical 

 halves of a uniform gravitating sphere. G. Greenhill adds 

 some remarks, ibid., 109, 1922, 778. One point that has 

 interested the writer of these notes is the following. It is 

 known that an ellipsoidal shell bounded by concentric similar 

 and similarly situated ellipsoids gives zero force inside. Is 

 the ellipsoidal shell the only one possessing this property ? 

 In terms of electricity the question is: a charged ellipsoidal 

 conductor, when isolated, has surface density proportional to 

 the perpendicular from the centre on the tangent plane : is 

 there any other form of conductor having this property ? 



Lambert's theorem in central orbits under the inverse 

 square law has recently been put into a form that leads to a 

 more extended usefulness of the theorem in astronomy 

 (M. Subbotin, M.N., R.A.S., Ixxxii, 1922, 383-90, 419-29). 

 The form is 



ITS* 



4<2 r + / 6^ 



where r, r^ are the hehocentric distances of a planet or comet 

 at two instants /, ^, a is the semi-major axis of the path, 

 s is the distance between the positions, and 6 is proportional 

 to t - /\ The quantity r is a function of the two quantities 

 (r + /)/4«2 and sI{y + /). The advantage of this form is 

 that T is in practice very nearly unity, and changes slowly. 



An article by A. R. Forsyth : " Differential Equations in 

 Mechanics and Physics" {Math. Gaz.,xi, 1922, 73-81), contains 

 many points of interest on the relations between pure and 

 applied mathematics. The author criticises present text- 

 books, where each writer develops the mathematics required 

 as if the methods and processes applied only to this subject, 



