AND PARTIAL DIFFERENTIAL RESOLVENTS. 53 



14. The values of V in (64), (65) which satisfy (50) 

 also, are the roots of the quartic 



V 4 +(a?+y-^)V a + *(a?-y)V + ^=o. . (66) 



Equation (66) will coincide with the quartic 



V 4 + «V a ±&V + c=o, .... (67) 



if x, y, z are determined from 



x + y — z i =a, (68) 



*(*-y)=+fl, (69) 



xy=-c (70) 



These values depend on the well-known cubic 



z 6 + 20s 4 + (a z — 4c)z z — b 7 - = o. 



See Todhunter's ' Theory of Equations/ p. 112, 2nd ed. 



Havant, Sept. 1881. 



Postscript. — Equation (6) is made linear by the relation 



4a 3 + 27m 2 _ cb , v 



** barrf — qmcF ' zdx i 



which gives 



d?z 



dx 1 



- log ( — — ; — - — T-)i- + 6«( , ° 2 2 = 0.(72) 



dx ° \ 2am 1 — 3mA 1 / dx \ 4a l + 2jm 2 J 



