AND PARTIAL DIFFERENTIAL RESOLVENTS. 51 



Differentiate (49) with respect to x, y, z respectively, 

 then 



(4V^ 2 ,Y + E)£ + |^ + f.y + f=o,.( 5 o) 



(4V^2,V + R)f + |.V^f.V + f=o,.(5i) 



( 4 V* + 2tY+TL) € ^ +$ .V z + d — .V + ^ = o. . (52) 

 dz dz dz dz 



Prom (50) and (51), from (51) and (52) there results 



\dy ' dy ' dy) dx \dx ' dx' dx) dy 



\dy dy' dy) dz \dz' dz ' dz ) dy 



Equation (50) is the first partial differential resolvent of 

 the quartic (49) together with the two conditional equa- 

 tions (53) and (54). 



The value of V, in functions of x, y, z 3 which satisfies 

 (50) _, and satisfies also the conditional equations (53), (54), 

 is a root of the quartic (49); and a root of the quartic (49) 

 is a solution of each of the equations 50 to 54. 



13. With a view, therefore, to obtain a solution of (53), 

 (54), it will be necessary to try a few simple assumptions 

 of the forms of t, R, S, which are suggested by the equa- 

 tions themselves. Put 



*=i- -- dR d ^ - 



dy dx dy dx ' 



By integrating these equations, then 



t=x + y + z 1} (56) 



R=(y-*)*„ (57) 



where z VJ z % are functions of z only. 



e2 



