220 Dr Baker, On the differential equations 



addendum a sketch of a proof of the theorem as to functions of 

 one variable possessing an algebraical addition theorem. This 

 theorem is that an algebraic simply transitive group of com- 

 mutative transformations is necessarily expressible by Abelian 

 functions ; the group is therefore associable with an integral 

 function of the canonical variables, this integral function satisfy- 

 ing in the case of one and two variables the differential equations 

 now under discussion. The question suggests itself whether 

 every simply transitive algebraic group is similarly derivable 

 from a single function of the canonical variables of the group. 



The variables u 1} u 2t ... being, as originally defined from the 

 hyperelliptic construct y 2 =f(x), expressible by sums of p integrals 

 respectively of the forms 



f dx C xdx 



J7f(w)' JvTp""' 



it is manifest that a linear transformation for x, say 



x = (az + b)/(cz + d), 



enables us to write u lf u 2 , ... as linear functions with constant 

 coefficients of variables v 1} v 2 , ... which have precisely similar 

 expressions in terms of integrals in which x is replaced by z 

 and the coefficients in the polynomial (f> (z) which replaces f(x) 

 are certain linear functions of the coefficients in f(x). The most 

 general linear transformation for x is however capable of being 

 built up from three particular transformations of the respective 

 forms x = mz, x = sr\ x = z + h. The first of these gives rise to 

 the isobaric character of the differential equations ; in any one of 

 the differential equations the sum of the suffixes of the coefficients 

 \ , \, ... and the functions ^ or p m entering, is the same for 

 each term. The second elementary transformation corresponds 

 to the property of the differential equations that they can be 

 arranged in pairs transformable into one another by altering \ i} 

 @ij, fPijui respectively into \ 2p+ ^i, tfp-i+^p-j+i^p-i+^p-j+hp-k+hp-i+i- 

 As to the third elementary transformation x = z + h, by taking h 

 small we find at once that it gives the fact that the system of 

 differential equations remains unaltered when acted on by a 

 certain operator, to be presently put down, provided that at the 

 same time certain suitable constants are added to the functions %>ij 

 in the new forms of the differential equations. In fact the simple 

 forms found for the differential equations are partly due to a 

 particular choice for a polynomial f(x, z) {Acta Math. xxvu. 



