1 872.] Mr. W. II. L. Russell on Linear Differential Equations. 15 



supposing them also entire functions of (a?). At present we shall confine 

 ourselves to the linear differential equation of the second order, 



ax ax 



The most general form of solution which this equation will admit will be 

 y={X + * s /Yy i . For if w © put y={X we should have three 

 particular integrals corresponding to the three cube roots of T, n being 

 supposed fractional. 



Substituting in the proposed equation, we shall find, after making some 

 reductions, 



n ( n _l)P{X' + iT-^T'} 2 + HP{X+Y i }{X'' + iY-^T"-iT-lT' 2 } 



+ nQ{X + T^}{X' + iT-Y'}+R{X+TH 2 =0; . . . . (1) 



from whence we have 



n(n— 1)PYX' 2 + in(n— 1)PY' 2 + nPXYX" + JwPYY" - | PY' 2 



+ nQXYX' + inQYY' + EYX 2 + EY 2 =0, (2) 



n(n- 1)PX'Y'Y+ ^iPXYY"- |nPXY' 2 + *PX"Y 2 



+ ^QXYY' + 7iQX'Y 2 + 2RXY 2 = (3) 



Let 



~P=a+Px + yx 2 + lv 3 , Q = a' + (2'x + y'x 2 , E=a"+/3"^. 

 This will give us 



JL=p-{-qx, Y=r-\-x. 



For if we assumed either X or Y to be of higher dimensions, we 

 should obtain, on equating the coefficients of the powers of (x) in 2 and 3 

 to zero, more equations than the total number of constants in the dif- 

 ferential equation and assumed solution. The number of disposable 

 constants is eleven. If we assumed X of two dimensions or Y of two 

 dimensions, we should have more than eleven equations. 



Now equate the coefficient of the highest power of (x) in (2) to zero, 

 and we have 



n 2 2+?i( y '-B) + /r=0, 



which determines n independently of the constants in X and Y. 

 Similarly, from (3), 



n (n _ a + & + n y ' + 2 t 3" = 0. 



This of course becomes an equation of condition between the coefficients 

 of the given differential equation, when we substitute the value of (n) 

 we have found from the last equation. 



To determine p, q, r, we proceed as follows : — In equation (3) put x 

 successively equal to the three roots of the equation 



Then we have three equations, linear in ^ r, r, from which these quan- 



