4.02 Prof. A. Gray on the Attraction of 



22. Returning now to the results expressed by equations 

 (16), (20), with lower limit X or 0, as the case may be, it is 

 obvious that we can change the lower limit of the integral. 

 Thus, taking (16) we write 



M + V x = w' + V 



where V, is positive as well as X. Then when u is equal to X', 

 u' is equal to V 1? and the equation becomes 



which is of the same form as (16) and the accents on the it's 

 may be omitted. This proves that two confocal homoeoids of 

 equal mass produce the same potential at any point P external 

 to both, that is, that their external fields are identical. If 

 the masses are different, the potentials (and therefore also the 

 field-intensities) at the different points are proportional to 

 the masses. 



This of course is a particular case of the very general 

 theorem of the potentials of distributions which was given by 

 George Green, of Nottingham, in his celebrated " Essay on 

 the Application of Mathematical Analysis to the Theories of 

 Electricity and Magnetism," published by subscription at 

 Nottingham in 1828. 



The substitution used above is also applicable to (20), and 

 proves that any two confocal solid ellipsoids of equal mass pro- 

 duce the same potential at every point external to both. If the 

 masses are different in the two ellipsoids, the potentials at the 

 same external point are proportional to the masses. This is 

 what is usually called Madam-in's * theorem ; but it was 

 only given in its full generality by Laplace many years after 

 Maclaurin's death. It is stated in Maclaurin's 'Treatise of 

 Fluxions ' (Edinburgh, 1742), § 653. that the attractions of 

 two confocal ellipsoids are the same at all external points 

 which are on the prolongation of the axes. This was a very 

 remarkable result for the time, and though the theorem ^vas 

 only fully generalised by Laplace in his book entitled ' Theorie 

 du Mouvement et de la Figure Elliptique des Planetes' 



* Colin Maclaurin, 1698-1746, Professor of Mathematics in the Uni- 

 versity of Edinburgh, appointed as assistant and (apparently) successor 

 to James Gregory in 1725. There being a difficulty, through want of 

 funds, in maliiDg this arrangement, Xewton offered to pay £20 a year if 

 Maclaurin were appointed. Thus Maclaurin was appointed, Newtono 

 suadente, as stated in the inscription on his monument. 



