382 PHILOSOPHICAL TRANSACTIONS. [aNNO J 7/8, 



account; and therefore the nice agreeement of Mr. Smeaton's experiments with 

 the theory cannot fail to add fresh evidence to these established laws of nature. 



I shall conclude these remarks with observing that since it is perhaps impossible 

 to give one general answer to all the arguments which are brought in favour of 

 the new doctrine of forces, it seemed very desirable that we should have a general 

 rule to direct us in judging of the cases that occur in practice. It is of more 

 consequence to the improvement of science and the good of the public, to 

 point out the source of mistakes, and the wisest means of avoiding them for the 

 future, than merely to confute and silence our adversaries. Some writers have 

 considered this question as entirely verbal, and have afFected to treat the advo- 

 cates on both sides with the greatest contempt. Such persons save themselves a 

 great deal of trouble, and have the credit of seeing further into the con- 

 troversy than others; but after all, I am afraid the practical mechanic will receive 

 little information or security from such speculations. Propriety of expression in 

 these matters is not all we want. When a [)lan is proposed for execution, and 

 a certain effect predicted, the grand object is, how to form a sure judgment 

 before-hand of the event, in order to prevent unnecessary expences; and I shall 

 think my time v/ell employed, if these considerations appear to have the least 

 tendency to promote so useful an end, in the opinion of that society to whose 

 learned and zealous endeavours we owe the very first important discoveries in the 

 year 1668, concerning the collisions of bodies. 



X^'in. Observations on tlie Limits oj' Al°chruical tlquations; and a General 

 Demonstration of Des Cartes's Rule fir finding their Number of x'lffirmalive 

 and Negative Roots. By the Rev. Isaac Milner, M. A., Fel. of Queen's Col., 

 Cambridge, p. 380. 



^ 1. The investigations of the limits of equations is considered as one of the 

 most important problems in algebra. The knowledge of them not only enables 

 us to demonstrate many useful theorems in that science, but is also of material 

 service in discovering the roots themselves. Mr. Maclaurin has treated this 

 subject very fully, both in his Algebra and in the Philosophical Transactions. 

 The substance of what he has delivered may be briefly expressed in the two 

 following pro|)ositions. 1st. That any equation x" — /jx""' -(- qx"-' — &c. = o 

 being proposed, if you take the fluxion of this equation, and ilivide it by ,f, the 

 resulting equation will have all its roots limits of the roots of the given equation, 

 idly. If the terms of the proposed equation be multiplied into the terms of any 

 arithmetical series, the resulting equation will also have its roots lin)its of the 

 roots of the original equation. 



§ 'I. This 2d proposition, though admitted by all the eminent authors, cer- 



