420 [Jan. 22, 



I have not yet analysed this new compound ; but both mode of 

 formation and properties (it crystallizes in white very fusible needles, 

 possessing the odour of creosote) leave no doubt that it is the 

 alcohol of the naphthaline series which has so long eluded the 

 researches of chemists. 



III. "On the Differential Equations of Dynamics. A sequel 

 to a Paper on Simultaneous Differential Equations." By 

 GEORGE BOOLE, F.R.S., Professor of Mathematics in 

 Queen's College, Cork. Received December 22, 1862. 



(Abstract.) 



Jacobi in a posthumous memoir*, which has only this year ap- 

 peared, has developed two remarkable methods (agreeing in their 

 general character, but differing in details) of solving non-linear partial 

 differential equations of the first order, and has applied them in con- 

 nexion with that theory of the differential equations of dynamics 

 which was established by Sir W. R. Hamilton in the ' Philosophical 

 Transactions ' for 1834-35. The knowledge, indeed, that the solu- 

 tion of the equation of a dynamical problem is involved in the dis- 

 covery of a single central function, defined by a single partial differ- 

 ential equation of the first order, does not appear to have been 

 hitherto (perhaps it will never be) very fruitful in practical results. 

 But in the order of those speculative truths which enable us to per- 

 ceive unity where it was unperceived before, its place is a high and 

 enduring one. 



Given a system of dynamical equations, it is possible, as Jacobi 

 had shown, to construct a partial differential equation such that from 

 any complete primitive of that equation, i. e. from any solution of it 

 involving a number of constants equal to the number of the inde- 

 pendent variables, all the integrals of the dynamical equation can be 

 deduced by processes of differentiation. Hitherto, however, the 

 discovery of the complete primitive of a partial differential equation 

 has been supposed to require a previous knowledge of the integrals 

 of a certain auxiliary system of ordinary differential equations ; and 



* Nova methodus aequationes differentiates partiales primi ordinis inter nume- 

 rum variabilium quemcunque propositas integrandi. (Crelle's Journal, vol. Ix. 

 p.l.) 



