831 



distant, yielded, as already remarked, only *4< per 10,000, part of 

 which was sulphate, part carbonate of lime. 



By certain management, I am informed, as by not allowing the 

 sea-water in the boilers to be concentrated beyond a certain degree, 

 the incrustation, in the instances of the transatlantic steamers, is in a 

 great measure prevented. Perhaps it might be prevented altogether, 

 were sea- water never used but with this precaution, and taken up at 

 a good distance from land, and in situations where it is known that 

 the proportion of sulphate of lime is small. If this suggestion be of 

 any worth, further, more extensive and exact inquiry will be re- 

 quisite to determine the proportion of sulphate of lime in different 

 parts of the ocean, and more especially towards land. By the aid of 

 the transatlantic steam navigation companies means for such an in- 

 quiry may easily be obtained ; and it can hardly be doubted that 

 the results will amply repay any cost or trouble incurred. 



Lesketh How, Ambleside, 

 March 29, 1849. 



2. "On the Universal Law of Attraction, including that of Gra- 

 vitation, as a particular case of approximation deducible from the 

 principle that equal and similar particles of matter move similarly, 

 relatively to each other." By John Kinnersley Smythies, Esq. Com- 

 municated by T. F. Ellis, Esq., F.R.S. 



After stating the general object of his investigations and explain- 

 ing the notation he employs, the author enters upon some prelimi- 

 nary geometrical inquiries. He gives the equation between the six 

 right lines drawn between four points in a plane ; the solidity of a 

 tetrahedron in terms of its edges : the equation between the cosines 

 of the six angles made by four right lines meeting in a point ; and 

 the equation between ten right lines drawn between five points, with 

 some formulae of verification. Giving some general rules for the 

 transformation and consolidation of series, he transforms the last 

 equation into one involving the solidities of tetrahedrons, and shows 

 how the sign of each tetrahedron in that equation is determined by 

 its position relatively to the least solid including them all ; and then 

 gives the equation between all the right lines drawn between n 

 points. 



Having shown that the result of differentiating the product of n 

 variables, m times successively may be derived from the mih power 

 of the sum of the n variables, developed by the polynomial theorem 

 by substituting for every power of each variable its differential of an 

 order numerically the same as the power; and applied the theorem 

 to find the differential of the mih order of the equation between ten 

 right lines drawn between five points ; the author gives the first four 

 successive differentials of the same equation in another form. 



Proceeding with his investigation he deduces the necessary equa- 

 tion between the distances and central forces of five moving points, 

 and derives from it the general system of equations which determine 

 the motion of any number of spheres in terms of <p (the function of 

 the distance according to which the attractive force varies), their 



