b 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 269 



The equation in 6 is 



which may be written /(-D)y=X; 



and y={/(-D)}-i . o + {/(-D)}-i . X 



the values of w being determined by the equation / (w)=0. 



Ex. 6. («. + /3r^+a(«. + /3r-* ^^+&c. = X. 



het af=:ax + 8, then ^ = a'" '^'^ ■^ (Part 1, Art. 27.) 



&c. = &c. 



a X — —+ aa ^ x' ^ + &c. = X' 



which coincides with Example 5. 



Ex.7. ^_«.i^_i^ = 



ax d ^i 1 X 



By multipljdng by x and reducing to differentials in 6, we get 



^^yt«^^(-l)-^^y + ^^=0 



/_D+i i\ jPdTiI „ 



or (_D + i)j, + a(-l)*/£i|e^y = 



or „ + a(-l)iL£±3e^y=0 



y+«(-i)*^^^5=o 



or ■Z,,(-i)-^^ W-D-^ JL^O 



^ ^ ^ /-D ^ 



or -^ a . ^ "" zrO. 



VOL. XVI. PART III. 3 Y 



