6 REPORT 1865. 



interesting to myself, and more justice would have been done to the writers whose 

 names have been little more than mentioned, if I could have completed the outline, 

 or, better still, have filled in the details. As it is, some apology is due for having 

 so long detained you upon mathematics ; hut, as a science whose rules all others 

 must obey, it has large claims upon our attention ; and if a personal motive must 

 also be confessed, one's mind lingers upon a favourite subject. 



Mathematics. 



On Dual Arithmetic. By 0. Byrne. 



The author explained his method of dual arithmetic, which he has applied, in 

 connexion with the calculus of form, to investigate the relations and properties of 

 angular magnitudes aud functions, plane and spherical trigonometry, &c 



On certain Theorems in Laplace's Discussion of the Figure of the Earth unit 

 Precession and Mutation. By Prof. A. H. Curtis. 



On the Theory of Differential Resolvents. By the Rev. 11. Haeley, F.ll.S. 



The theory of differential resolvents owes its origin to the discovery that, from 

 any algebraic equation of the degree n, whereof the coefficients arc functions 

 of a variable, there may be derived a linear differential equation of the order «■— 1, 

 which will be satisfied by any one of the roots of the given algebraic equation. 

 These differential equations are now known by the name " differential resolvents." 

 The author explained how they are formed, and pointed out their connexion with 

 the theory of algebraic equations. 



One of the most important of his recent results is the following : — 

 If u represent the with power of any root of the algebraic equation 



y n -m / "- r +(n-l).r = 0. 

 then u, considered as a function of x, satisfies the linear differential equation 

 -Fn—r d m~\*— r r rf"l' , i\T« d m -,"1" . 



in which the usual factorial notation 



[«]»=(«) (a-l)...(«-&+l) 



is adopted. And the complete integral of this differential equation is 



w = C 1? /,- + <', //./<<... +C,,y„<<<, 



y v ji,, . . . y n being the n roots of the given algebraic equation. From this theorem, 

 which is an extension of one given by the late Prof. Boole in the Philosophical 

 Transactions for 1804, p. 735, all the differential resolvents of algebraic equations of 

 the above trinomial form may be readily deduced, by making m=l, in which ease 

 u=y, and depressing the differential equation by immediate integration. 



On Chasles's Method of Characteristics. By Professor T. A. Hirst, F.li.S. 



Alter briefly explaining the nature and scope of this important method, by which 

 the theory of conic sections has now been completed, the author communicated a 

 few of the results of Professor Chasles's most recent researches on the properties of 

 cnnics in space, which satisfy one less than the number (eight) of conditions neces- 

 sary to determine them. These results were communicated to the Academy of 

 Sciences, on September 4, 1865, and* appear in the ' Comptes Rendus ' of that date. 



On Quadric Transformation. By Professor T. A. Hirst, F.li.S. 

 The object of the paper was to establish new properties of two figures (in one and 

 the same plane) so related to one another that to a point in one figure corresponds 

 but one point in the other, and rice rcrxa, whilst to a right line in the one figure 

 corresponds a, conic section in the other. Among these properties were several 

 which exhibit a remarkable connexion between a correspondence of this kind and 



