184 [June 19, 



grounds of the method, it is a perfect method of interpolation. The 

 analytical investigation, however, shows that, for the mere purpose of 

 interpolation, the process might be modified by altering the coefficients 

 of V without affecting its form ; but it indicates at the same time that 

 such modifications have no definite analogy with that process by which 

 weight is assigned to astronomical observations, and, from their arbi- 

 trary character, lead to results which cannot properly be regarded as 

 expressions of probability in any sense. 



XVI. " On Simultaneous Differential Equations in which the 

 number of Variables exceeds by more than unity the number 

 of the Equations." By GEORGE BOOLE, Esq., F.R.S. 

 Received June 19, 1862. 



(Abstract.) 



This paper contains the proof, with some applications, of a method 

 described in a paper bearing nearly the same title which was pub- 

 lished in the * Proceedings of the Royal Society ' for March 6, 1862. 



XVII. " On the Calculus of Symbols." Third Memoir. By 

 W. H. L. RUSSELL, Esq., A.B. Communicated by Pro- 

 fessor STOKES, Sec. R.S. Received June 18, 1862. 



(Abstract.) 



The following paper is a continuation of the two preceding Memoirs 

 on the same subject. It has a fourfold object. In the first place, I 

 calculate the general values of the coefficients in the Binomial Theorem 

 given in the first Memoir. In the next place, I give an expression 

 for the form of the coefficient of the general term of the multinomial 

 theorem as previously explained. I then give a theorem for the 

 multiplication of symbolical factors emanating from each other after 

 a given law ; and lastly, I investigate a binomial theorem, reciprocal 

 to the binomial theorem already considered. 



