roots of 2n + J the algebraic sum of which roots is one, and dimi- 

 nishing the middle difference by theorem Q until it reaches a number 

 nearest to half the sum of the first and third differences. The 

 difference between 2+l and the number thus obtained will be the 

 double of a triangular number = 2T. By the next step, the extreme 

 differences are reduced until they are of the form m, m + 1 ; and the 

 difference between 2n -f 1 2T and the number thus obtained will 

 again be the double of a triangular number = 2T>. The differences 

 last obtained give the double of a triangular number + 1=2T" + 1. 

 So that we find 2 + 1 = 2T + 2T' + 2T" + 1 . Consequently n== the 

 sum of three triangular numbers, if all the three operations be 

 necessary; if not, to two or one triangular number only. 



II. The first part of a paper " On a Class of Differential Equa- 

 tions, including those which occur in Dynamical Problems." 

 By W. F. Donkin, M.A., F.R.S., F.R.A.S., Savilian Pro- 

 fessor of Astronomy in the University of Oxford. 



This paper is intended to contain a discussion of some properties 

 of a class of simultaneous differential equations of the first order, 

 including as a particular case the form (which again includes the 

 dynamical equations), 



i dZ * rfZ /T v 



*i=-r> y'i= -j-. (!) 



ay i dx i 



where x l ... x n , y t ... y are two sets of n variables each, and accents 

 denote total differentiation with respect to the independent variable t ; 

 Z being any function of x } &c., y x &c., which may also contain t ex- 

 plicitly. The part now laid before the Society is limited to the 

 consideration of the above form. 



After deducing from known properties of functional determinants 

 a general theorem to be used afterwards, the author establishes the 

 following propositions. 



If x l ... x n be n variables connected with n other variables y 1 ...y n 



/7\ 



by equations of the form y t = (X being a given function of 



(tXi 



#!...#); then the equations obtained by solving these algebrai- 



