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PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 265 



dx" \2x to^s-'mJ ^^.n-l 2.4 x^ ^,^n-2 + ^'^- 



+ (_.-|)"-l 1-1 -3 (2n-3) n{n-l) 1 _^ X 



2 .4.6 ... .2n • x" "~ m^a^n 



CoE. 1. If n=l, the equation for determining u is 



du_ /Jl l_v X 



dx \2x m^x^J 



u = —- 



in" X 



1 1 



of which the solution is u=- ^ '" '^ / f^jL^<^2;= 



1 



l_ _j_ 1 



and «=(l + »«T— t) (4l! -? / g Xrfg X 



Cor. 2. If »=l, X= -r-, it is evident that 



where ^n is the solution of the equation without X (Ex. 3.) 



It appears, therefore, that the complete solution of equations of this form is 

 reduced to the solution of ordinary linear equations, and the determination of the 

 half differential coefficient of the results. 



Ex. 9. y-mx TT=X, where n and r are any whole numbers. 



d x' '*'' 



where v + w is the solution of the equation 



\ dx^-^^ dx^ + i/^ ' 



Now —7X - — -rr = ^ r-9irrr+ Kr + h)nx 



dx^ + i rf2:'" + i rfar^'^ + l ^ ■^ dx^' 



VOL. XVI. PART III. 3 X 



