224 



Dr Baker, On the differential equations 



isobarism, (ii) the condition of being alike in pairs when the 

 suffixes near their upper and lower values are interchanged, 

 (iii) the condition of allowing the operator above denoted by P ; 

 and in fact it is seen on examination that all the 15 equations 

 are derivable from two of them, namely from Q 2222 and Q nn , by 

 combinations of the two conditions (ii) and (iii). 



It follows of course in the same way that the \p (p — 1) 

 rational relations connecting the |p(p + l) functions (p^- allow 

 the three operators P, Q, R described above. Thus for instance 

 the sixteen-nodal quartic, for p — 2, 



G 3 -2y, 0,-lx, C l3 



G n 



G,-\z, C s + %y, G 2 + %x, C x 



G 5 , Ot + iz, G 3 + %y, (7,-f* 



Gq, 



G 5 , 



C 4 -%z, G 3 -2y 



= 



is isobaric, x, y, z, Gi being of respective weights 4, 3, 2, 6 — i, is 

 also symmetric in regard to x, y, z, C , C lt ... and z, y,x, G e , C 5 ,..., 

 and admits the operator 



9 ~ 9 ~ 9 ~.~ 9 



+3C 4, +4C 4 



+ iG 'dd+ 60 'dc, 



But on the other hand this surface also admits the group of 

 two parameters whose equations are given by the expressions for 

 #> 22 (u + a), p m (u + a), @ n (u + a) in terms of jp 22 (u), ® 21 (it), g> n (u), 

 g) 22 (a), fp 21 (a), p n (a). If a, b, c, d denote the minors of the elements 

 of any row of the determinant just written down, the generating 

 operators of this group are 



9,9 9 

 ox oy oz 



7 3 d j ° 

 ox oy oz 



and the finite equations could easily be found by differentiation 

 from the formula given in the American Journal (Vol. xx. 

 p. 381) for the function 



a (u + a) a (u — a) 

 a 2 (u) a- (a) 



in terms of the functions |j? 22 (iO> •••> fPn(a)- 



