132 Archdeacon Pratt on a Problem in 



gents, seems to corroborate these views. These agents, which 

 are known not to act upon benzol, ammonia, or water respec- 

 tively, leave the component groups of amido-benzol and hy- 

 droxyl-benzol unchanged. Thus it appears that their action 

 consists mainly in the abstraction of hydrogen from the methyle 

 group contained in toluidine (amido-methylbenzol) and cres- 

 sol (hydroxyl-methyl-benzol), thereby increasing the atomicity 

 of the residuary groups, and causing a coalescence of three phe- 

 nyle molecules. 



In the case of reagents being employed which at the same 

 time exert an action upon, or effect a substitution in the benzol, 

 ammonia, or water residues of aniline and phenol respectively, 

 we can easily recognize the formation of corresponding by- 

 products which interfere with the practical values of these pro- 

 cesses for the formation of colouring-matters. 



XVIII. To find what changes may be made in the arrangement of 

 the mass of a body, without altering its outward form , so as not 

 to affect the attraction of the whole on an external point. By 

 Archdeacon Pratt, F.R.S* 



1. T ET A represent the body in the first instance, and B 

 -I-J after a rearrangement of the mass has taken place. 

 Suppose each particle of B is subtracted from the correspond- 

 ingly situated particle in A ; we shall thus obtain an imaginary 

 body A — B, the density of some parts being positive and of other 

 parts negative, the whole mass being zero, and its attraction on 

 any external point zero. Our problem amounts to finding the 

 form and law of density of such a body. By following this pro- 

 cess we shall discover the changes we seek for, even if they ren- 

 der the lav/ of the density of the mass discontinuous. 



2. The surface must evidently be a closed one. Let the origin 

 of coordinates r, 9 } co be within the surface, p the density, r and 

 a the general and mean radii of the surface, 



r=a.2/ = a(l+w 1 + w 2 + w 3 H- . . .) 



the equation to the surface, — the first term in this series of La- 

 place's functions which represents u being 1, because a is the 

 mean radius. Suppose p is expanded in a series of powers of r, 

 and let 



,=F + G(0 +H QV..., 



where F, G, H . . . are functions of yu-(= cos 6) and co only. 



3. Let c be the distance of the attracted point from the origin, 



* Communicated by the Author. 



