May 20, 1922] 



NATURE 



647 



Now from the diagram it will be seen that /^CE'A 

 =--^CPE = <?, say, and EP = E'A = y. The attraction 

 due to E at P is equal to kvd^ cos 6, but this is 

 clearlv, if y = EP, 



k<y' 



a2 <iS' cos 9 



Now the factor dS' cos 6 jr^ is clearly the solid angle 

 subtended at A by the element dS'.- The whole force 

 exerted at P by the shell is thus, to a constant factor, 

 equal to the solid anglesubtendedat A by the whole con- 

 centric surface of radius /, which is ^ir. The attraction 

 of the shell on a unit particle at P is thus k^ircra^ip, 

 that is, it is the same as it would be if the whole 

 mass were collected at the centre. 



If the point P be internal to the shell the concentric 

 surface with A falls within, and the total solid angle 

 subtended by the shell at A is zero so tha't the attrac- 

 tion is zero. 



This process extended to an ellipsoid and the 

 confocal ellipsoid through an external point is made 

 to give the force due to the shell at the point. 

 The integration is made immediate by the use of a 

 theorem of solid geometry which holds, as I pointed 

 out, for confocal conicoids. The theorem may be 

 stated here. Let A and P, E and E' be pairs of 

 corresponding points ; then the distances AE' and 

 PE are equal, also if p and p' be the lengths of the 

 perpendiculars from the centre on P and E', the 

 angle which PE makes with the perpendicular p, 

 6' the angle which E'A makes with the perpendicular 

 p', then the theorem holds — 



P ^ P' 

 cos e cos d'' 



This theorem shows the result of the integration over 

 the ellipsoid to be, to a constant, equal to the solid 

 angle subtended at an internal point by a closed 

 surface in the manner just illustrated by the spherical 

 shell. It is curious that this geometrical theorem 

 which enables this result to be obtained is, as I have 

 found, generally unknown to writers on geometry, and 

 is not contained in any of the treatises which I have 

 examined. 



The next problem is one of which, I believe, the only 

 simple solution given before 1900, was due to the late 

 Prof. Tait, of Edinburgh. The problem was the deter- 

 mination of the pull between the two halves of a 

 homogeneous sphere due to gravitational attraction. 

 Prof. Tait's solution was a quasi-hydrostatic one, and 

 I believe that he held the opinion that the only choice 

 was between this and straightforward sextuple in- 

 tegration. There are, however, at least three other 

 methods of attacking the problem, and one of these 

 which occurred to me a long time ago I will indicate 

 here. This has only been published so far as 1 

 know in a collection of exercises lithographed nearly 

 twenty years ago by the late Dr. Walter Stewart, who 

 was then my assistant, for the use of students in 

 Glasgow. It makes use of the theorem of Gauss 

 referred to above. 



Consider the homogeneous sphere of radius a and let 

 a closed surface be described consisting of a plane 

 part dividing the sphere into two segments, and a 

 spherical part fitting close to the smaller segment of 

 the sphere. The surface integral of normal force over 

 this surface will consist of two parts, I, the integral 

 over the plane, and 2 the integral over the spherical 

 portion. The mass M of the enclosed segment can 

 easily be calculated and ^irkM is equal to I + - ; of 

 course - is also easily calculated, and thus I is obtained. 

 If r be the radius of the plane section, z the distance 

 of that section from the centre, p the density of the 

 sphere, the mass of unit area of a disc of radius r and 

 thickness dz is pdz. Multiplying this by I, we see that 



NO. 2742, VOL. 109] 



the product Ipdz is the force due to the whole sphere 

 on the disc of radius r and thickness dz, and if this be 

 integrated from z-a to ^ = d we obtain the attraction 

 of the whole sphere on the hemisphere throughout 

 which the integration has been carried ; this attraction 

 of the whole sphere on the hemisphere includes 

 the attraction of this hemisphere on itself, which, 

 of course, is zero. Thus the integration gives the 

 attraction of one hemisphere by the other. 



The mass M of the segment within the closed 

 surface is easily seen to be 



irp[2a^- -^a^z + z^) ; 



the integral of normal force over the curved part of 

 this segment is 



thus 



2 = 27rAa2/i-|\4^^p. 



l + ~kpTr'^a'^(i-^\ =4^p7r2(2a3-3a2^ + ^3 



that is 



l = ^kpirH{z^-a''). 



We have therefore for the product of I by the mass 

 per unit area of the disc coinciding with the plane 

 surface of the segment 



lpdz^^kp'^ir^z{z 



a^)dz. 



Integrating from z — a to z = o we get for the pull P 

 on one hemisphere exerted by the other, 



P = -A7rVaS 



or 3^iVP/I6a^ where M is the mass of the sphere 

 supposed of uniform density p. 



A numerical estimate of P for the earth must be 

 very rough, for the earth is not of uniform density, 

 and there are other causes of inexactitude. But by 

 the formula an estimate can be made in any units 

 that may be preferred. In c.g.s. units k is 6-yx 10-*. 

 The force between the two hemispheres of a body of 

 such great dimensions as the earth must be almost 

 entirely due to gravitational attraction (for cohesion 

 must be negligible in comparison), and this figure 

 may be taken as giving an idea of its amount. 



Andrew Gray. 



The University, Glasgow. 



The Conquest of Malaria. 



The obituary notice of Sir Patrick Manson, in 

 Nature of May 6, concludes with the hope that his 

 memory may ever be kept alive as the Father of 

 Tropical Medicine. As to this it is not difficult to 

 forecast that the medical profession will fully concur. 

 To the enthusiasm and inspiring teaching of Manson 

 is due the existence of tropical medicine as a speciality, 

 and the ever extending benefit tropical races receive 

 at the hands of men trained on the lines indicated by 

 him. 



In the present day, the views of the medical pro- 

 fession are apt to change rapidly in accord with 

 accumulated investigations and experiences of world- 

 wide origin ; opinions rigidly adhered to for fifty 

 years may be rendered taboo by a single telegram 

 received from some expert at a remote corner of the 

 earth. If the new view stands the test of criticism 

 the practical results are grasped ; but few care to 

 memorise how the change was effected. If this be so 

 with the profession specially concerned with disease 



