

1895.] On the Abelian System of Differential Equations. 301 



establishing the existence of a fluid interior, as supposed by M. 

 Folie, rather affords an additional reason for discarding this hypo- 

 thesis. 



II. "On the Abelian System of Differential Equations, and 

 their Rational and Integral Algebraic Integrals, with a 

 Discussion of the Periodicity of Abelian Functions." By 

 Rev. W. R. WESTROPP ROBERTS. Communicated by Rev. 

 G. SALMON, D.D., F.R.S. Received January 17, 1895. 



(Abstract.) 



Before entering on the discussion of the Abelian system of 

 differentia] equations, I treat of some general algebraic theorems 

 having reference to the differences of various sets of " facients," and 

 give a wider definition to the term " source," hitherto used to signify 

 the source of a covariant, and treat of two operators, and A. 



I then show how, by forming what I call a " square-matrix," all 

 the conditions can be obtained which are fulfilled when a polynomial 

 /(2) of the degree 2 n in z is a perfect square. With regard to these 

 conditions, I remark that any one of them being given all the others 

 can be found by successive operations of the operator 8. 



I next treat of the system of differential equations termed 

 "Abelian," in which there are m quantities and m 1 equations, 

 comprehended in the typical form 



z*dz 



S ^ = ' "',,.. 



where 2 relates to the m quantities z b z 2 , . . . . , g m , and i may have any 

 integer value from i = to i = m 2, it being understood that f(z} 

 is -a polynomial of the degree 2m in z ; and I show that, if 

 f(z)=z* m + P l z 2 l - 1 + P z z 2m - z + ____ P 2wt , be reduced to the degree 

 2m 2 in z in the following manner 



/ (*) + {0 0)} 2 -20 00 . L 00 = F (z), 

 where 



(z) EE (ZBI) (z z 2 ) ---- (z Zm) = z m +piz m ~ 1 + .... p m , 



T> 



and L (z) = z m -\ - z m ~ l + V w ~ 2 + X^" 3 +....+ \ m , 



2 



' ^z A*, . . . . , ^m being ra 1 arbitrary constants, all the rational and 

 integral algebraic integrals of the Abelian system 



z 2 



