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



We have seen that if Pe", when substituted for y, satisfies a linear 

 differential equation, w is an invariant, that is, it remains the same func- 

 tion of x, and the constants involved in the equation whatever numerical 

 value we are able to assign to these constants, on the supposition that P 

 and io are rational functions of (x). We shall call this the " principle of 

 index invariance," and proceed to show that it is true when w is irrational. 

 Let 



(a + j3x + y^ 2 )|| + (a + + y'v 2 + a VyJl + (a" + fi"x + y 'V + 3' 'x?)y == 



be a differential equation ; to find the condition that it may be satisfied by 

 , where w is irrational. The irrational quantity in w must be 

 of the form V X, otherwise the equation would admit of more than two 

 particular integrals. Moreover the factors of X must be factors of 

 a + fix -f yx 2 . We will suppose X= a + fix + yx 2 . Hence we may assume 



w=ju + yfl? + pV a + fix-\-yx 2 ; 



for if we expand w in descending powers of (x), and substitute the 

 resulting value of y in the differential equation, we shall easily see that 

 it can have no power higher than the first, fx, v, and p are of course con- 

 stants to be determined, and the double sign of the radical will give rise 

 to four equations instead of two, which will imply an equation of 

 condition. 



We now substitute y=¥ef uldx , where P is an algebraical function of (x), 

 in the linear differential equation, and thus obtain 



(a + fix+yX 2 ) | -^ + 2(^ + ^ + 10 V a + fix+yX 2 )-^ 



+ + vx + p s/a + (3x+yx 2 fV + + vx+p Va + j&r+y^yP j 



+ (a' + /3^+ 7 V + aV) | ^ + (/* + vx + p V a +fix + yx 2 )V } 



+ (a" + /3> + 7' v + a' V)P = o. 



The coefficients of the two highest powers of (x) must be equated to 

 zero, that is to say, the coefficients of the two highest powers of the 

 multiplier of P must be equated to zero. 



The following terms of that multiplier will be sufficient for our 

 purpose : — 



a' V + (yx 2 + g V)(ju + VCC + p ^a + @X+yX 2 ) 

 4- (fioG + yX 2 ){fx + VX + p ^ a +(3x-\-yX 2 f, 



or 



a' v + ( y v + a + +pWy(i+^+. .)) + O + r* 2 ) 



