514 Mr Baker, On a certain system of [May 2, 



expressing that partial differential coefficients, of the dependent 

 variable w, of the fourth order, are rational functions of the 

 partial differential coefficients of the second order, neither the 

 partial differential coefficients of other orders, nor the independent 

 variables u 1} u 2 ,..., entering explicitly. For definiteness suppose 

 there are three independent variables ; then the greatest possible 

 number of equations is fifteen, and the rational functions contain 

 six partial differential coefficients of the second order. If we 

 differentiate the equations once, and eliminate the resulting 

 differential coefficients of the fifth order between pairs of the 

 equations, we shall obtain at most twenty-four equations, each 

 linear in the ten partial differential coefficients of the third order, 

 the coefficients of these being rational in the differential coeffi- 

 cients of the second order. The elimination of the differential 

 coefficients of the third order, which we suppose to be possible, 

 will then lead to rational relations connecting the differential 

 coefficients of the second order. We suppose that these rational 

 relations are algebraically equivalent to as many as there are 

 independent variables, namely three, which we may denote by 



/i O, V, *, £ V, = 0, / 2 O, y, z, £ 77, £) = 0, f 3 (x, y, z, £, rj, £) = 0, 



where x, y, z, £, 77, £ denote the partial differential coefficients of 

 the second order, 



_ d 2 w d' 2 w d 2 w t d 2 w d 2 w ., _ d 2 w 



du 3 2 ' du s du 2 ' du 3 duj' du 2 2 ' di-i^du!* duj 2 ' 



In the course of deducing these three rational equations we 

 shall obtain such equations as 



1 d 3 w 1 d 3 w 1 d s w 



P dvJ Q du 3 du 2 R 3m 3 2 3m 1 



1 d 3 w 1 d 3 w 1 d 3 w 



P 2 du 3 du» 2 Qj du 3 duodu x R 1 dugdu^ 



where P, Q,..., R 1 ,..., are definite rational functions of x, y, z, 

 %, 7), £ Putting each of the fractions equal to /x we have 



9 4 w d 3m ,/3P d dP n 3P D 3P D dP n 3P D \ 



du 3 3 du 2 du 3 \dx dy 6z 9£ dv c d% 



hence, replacing the left sides by their rational expressions 

 R3333 (p, y, z, £, v, 0> P3332 (pa, y, z, £ v, 0, and eliminating ^ , we 

 obtain fi 2 as a rational function of x, y, z, £, rj, £ — an d can hence 



