On Solids of greatest Attraction, or Repulsion. 269 

 Therefore, C representing a constant quantity, the ex- 

 pression rrr5:i!^Lyi^^±yi^ + cf//A{x,y,z)^^ 



is to be a maximum ; and, by the method of variations, we 

 find, for the equation of the superficies, ■ „^\^.,h " "** 

 C.J(x,2/,.)=0, or-^^^^^ + C = (1.) 



Cor. 1. We learn, from the preceding analysis, that the 

 ficrure of the solid is independent of the law of density, l» 

 all cases. This leads me to observe, that although Mr. 

 Playfair, in the Edinburgh Transactions, restricts his pro- 

 blem to the case of homogeneity, there appears to be no- 

 thintr in his ingenious manner of treating the subject, 

 whidi renders such a supposition necessary. For, when 

 wp are directed to take a small portion of matter from a 

 point at C, and place it at another point D, it may be con- 

 ceived to be contracted or dilated, at this latter point, as 

 any variable law of density may require, without making 

 any difference in the reasoning, or in the result. 



'Cor. 2. If we suppose (p{x,7j, z) to be a function ot the 

 distance, or of the form ^{x--{-y'-¥z-), equation (l) takes 

 the form a:=^^(T^ + y^ + ^0, ^hich is the general equation 

 of solids of revolution: see ^ion^e «« Application, &c. 

 p. 18. 



Scholium. 

 Although it appears from what has been shown,' that 

 when the force is a function of the distance, the solid of 

 greatest attraction must be a solid of revolution; yet the 

 converse is by no means true ;— that the particles must ne- 

 cessarily act with a force, which" is some function of the 

 distance, in order that the solid of greatest attraction may 

 be a solid of revolution. Thus, if we want the figur e 



of the solid when the force, or <S^{x,y,z) = 1^:^ t 



X 



equation (l) becomes j^-^ + C =0, or x* + C(ix— 



d — • 



X 



y-^ — z^) = 0, evidently the equation of a solid of revolution 

 round the axis of x. Let a be the value of x when y and 

 x-0, then we have C = ^, and by substituting this 



value 



