AND PARTIAL DIFFERENTIAL RESOLVENTS. 47 



then 



(m 1 ) 2, _ r z 

 4« 3 + 2jm z ~~ 27' 



Hence there results by substitution 

 d z y dr dy r 



dx z rdx dx 9 



y = o, . . . (25) 



which is the second differential resolvent of the cubic 



f+ay+— %}€** (26) 



The value of (m) is found by integrating (24) and solving 

 algebraically with respect to (m), to be 



m=-e-f rdx — — e frdx (27) 



2 27 v " 



7. If y and z are such as to satisfy the equation 



dy dr . ~ 



ydx irdx* * 



substitute this value of y in terms of z in (25), and it 

 becomes 



£+^=i{- 2 i(i) + © i+ f }'■ (29) 



an equation which is soluble by means of (26) and (28) 

 for all values of the function r. 



8. Put r=Ax zn , where (A) is constant, then 



dz n % n(n+i) A z ,„ , x 



S + "^=-W + 9^' • • • (30) 



which is soluble by means of (31) and (32) :— 



