1862.] 13 



to this substance, it is accompanied by several other bases, the study 

 of which is not yet completed. Nor am I at present in a position to 

 offer any definite opinion regarding the constitution of the new com- 

 pounds, tempting though it appears to venture on speculations. It 

 is in the hope of rendering the formulse of the new bases more trans- 

 parent that I have commenced to examine some of the products of 

 decomposition. Their study is likewise far from being completed ; 

 but I may mention, even now, that both rosaniline and leucaniline, 

 when in nitric solution, are powerfully acted upon by nitrous acid, 

 new bases being thus generated, the platinum-salts of which are 

 remarkable for their fulminating properties. A splendid crystalline 

 base also deserves to be mentioned, which, associated with aniline, 

 appears among the products of distillation of rosaniline. 



The results obtained in the further prosecution of these studies I 

 propose to lay before the Royal Society in a future communication. 



II. "On the Integration of Simultaneous Differential Equa- 

 tions." By GEORGE BOOLE, Esq. Received March 4, 

 1862. 



It is well known that a system of n I simultaneous differential 

 equations of the first order connecting n variables always admits of 

 n I integrals, each of which is the form P=c, i. e. each of them is 

 expressible by a function of the variables, equated to an arbitrary 

 constant. 



But when the number of the variables exceeds by more than by 

 unity the number of the differential equations, no existing theory 

 assigns the number of theoretically possible integrals, or guides us 

 to their discovery. 



Yet cases such as this occur in problems of the greatest import- 

 ance. The solution of partial differential equations of the second 

 order by Monge's method depends ultimately on the solution of a 

 system of three ordinary differential equations of the first order 

 between Jive variables. 



I wish here briefly to indicate the results of a theory which enables 

 us in all such cases, 1st, to assign ti priori the number of possible 

 integrals ; 2udly, to reduce the determination of the integrals to the 

 solution of a system of differential equations equal in number to the 



