1872.] Mr. W. H.L. Russell on Linear Differential Equations. 17 



Then r must be one of the quantities fi v ju 2 , jj 3 ; we will suppose p v 

 Moreover, we must have 



or 



whence 



Generally, if 



1*3 — f*2 



dx 11 doc 91 ' 



be a linear equation, admitting a solution of the form 



every quantity which makes X + v^Y+x/Z— . . .to vanish will make 

 P vanish; every quantity which makes T +^/Z + . . . to vanish will 

 make P vanish; every quantity which makes Z-f. . .to vanish will 

 make P vanish. This principle is of course of the greatest use in de- 

 termining the form of this function. 



The most general form of irrational solution for a linear differential 

 equation of the third order will be {X + VY}' 4 or {X+^/Y} n ; for one 

 of the fourth order we shall have 



{X+ Vf}», {X+^Y}», {X+ VT+ vZy, {X+ Vy+VZ}", 



where (n) is supposed to be fractional. 



In all these cases the determination of (n) will be independent of the 

 constants in X, Y, Z. Por if we expand the proposed forms of solution 

 in descending powers of (00), substitute in the differential equation, and 

 equate the coefficient of the highest power of (x) to zero, we obtain an 

 equation to determine (n) not involving these constants. 



If X, Y, Z, . . . are rational fractions, this method will not apply ; but 

 since the factors of the denominators of these fractions, not regarding 

 their powers, must also be factors of P, we may proceed as follows : — 

 Let 00 — a =z be one of the factors of P, substitute oo=a-\-z in the given 

 differential equation, and let — m be the greatest negative value of 

 obtained by substituting y— z^-\- B# +1 ... in the differential equation, 

 and equating the lowest term of the result to zero. Again, let x—b=z, 

 and let — r be the greatest negative value of jjl, obtained by substituting 

 x~h-\-z in like manner. Then, if we put v = (x — a) m (x— b) r ... we 

 may obtain an equation whose irrational solution will be free from nega- 

 tive factors. 



VOL. XXI. 



