16 Mr. W. H. L. Russell on Linear Differential Equations. [Nov. 21 , 



tities may be determined. But putting x=0 iu equation (1), the equa- 

 tion becomes 



/ i \ 2 i t , n a'r , ,, 2 . n 2 a na , 2 n 

 n(n—l)aq 2 r+na rjpq+-— + a rp 2 + ———. + a r =0. 



Hence, substituting for ^ and r the values we have just obtained, we have 



a linear equation to determine p 2 ; hence p, q, r are determined. 

 Now, suppose P, Q, R to be such that we may assume 



X —p + qx + x 2 , Y = m + rx -J- sx 2 ; 



then the terms of equation (2), which contain the two highest powers of 

 x, are linear in T, or we may divide it out in equating their coefficients to 

 zero. Hence we may determine q at once. 



Put for x successively in (2) four roots of the equation P=0 ; then we 

 have four equations linear in p 2 , p, m, r, s, whence we have a quadratic 

 equation to determine p, and then m, r, s are known. "We have beside 

 a number of equations of condition. 



Now suppose P, Q, B. to be such that we may have 



lL=p + qx-\~a?, Y=r+sx ; 

 then we shall find that the terms which contained the three highest 

 powers of (x) are linear in T ; consequently p, q, and therefore X, may 

 be determined at once. X being known, we may obtain a series of 

 equations easily by which the constants in T may be determined*. But 

 we may adopt a somewhat different method, which will be useful in 

 many cases of this nature. Resuming the equation 



equation (1) shows that a + fix + yx 2 + must vanish for every quan- 

 tity which causes X+ \/Y to vanish, and equation (3) that the same 

 quantity must also vanish for every quantity which causes Y to vanish. 

 Let 



a + fix + yx 2 + &*? 3 = Kf*i + #)(/* a + 00 3 + °°)' 



* If 



Y = q +q 1 x+q 2 x 2 +- . . . +q^, 

 the terms which contain the fx highest powers are linear in Y ; and therefore 



PiP* • • • Pfi-l 



can be determined at once, commencing with^> _ 1 - 

 If M 



the terms which contain the (fi + l) highest powers are linear in Y; and therefore 



PoPi ■ ■ • ■ P^-i 

 may be determined at once. — W. H. L. R., Not. 21, 1872. 



