SATISFYING LINEAE DIFFERENTIAL EQUATIONS. 35 



The last gives - ,', (0s 1 ) 3 - 3 32 = 0, so that 6*3!= - 3(rc 32 )S, where any cube root may be taken, the other 

 roots giving # 3 2 , 3 3 . 

 Further 



3z 4 2 - 3 W - Wx - (1 + <V) .-Ci 1 = 0, 



3/4 3 - flsW - Wtf + (1 - 01 1 ) Zj2 = 0, 



- tfsW - ^2 S + (3 - 6V) ;,8 - SasaKi 1 - SoziX; 8 = ; 



of which the last gives 



-fcV.fW-aaieV-o, 



so that ftj 1 = 2i/(8o)J, and #.r, 0o 3 are given by taking the other roots for ( 32 )*. 



Lastly the equation 



- # 3 W - foxa 3 + (2 - 6V) z 2 3 - 3a 82 a; 3 i - 3a. n xJ = 

 gives 



,3 



which gives 



so that 6' 1 1 = 5. Similarly, 0, !i = 1 8 = 5, and a subnormal form exists satisfying the equation, of which the 



>'/ _ _f^]_ 



first column has the determining factor c 2 -' " ;E '' tS, and the 'other columns have the same factor with the 

 other cube roots of so. 



We may remark that this agrees with the results obtained for this equation by the ordinary methods 

 (FoRSYTH, "Linear Differential Equations," 99) under the assumption that 32 =^0. We have shown 

 that this is a necessary condition for the existence of the subnormal form in the variable /- ,~S satisfying 

 the equation formally, unless we have also <i 21 = 0. 



If, however, 32 = and n^i^O, then, as we have seen above, the transformation / = il will give us a 

 system admitting of 3 normal solutions; the equation for 0\ is, in fact, (#i) 3 - 4 2 ]0i = 0, giving } =0 or 

 2rt 21 *. 



We see, in fact, that, when (130 = and ff-Ji^O, the characteristic index of the original equation is 2, so 

 that there will be one regular form satisfying the equation, i.e., an expression of the form ./- P P (./ X 



If a 3 v, a 2 i ar c both zero, the equation is of Fuchsian type. Thus the normal or subnormal forms are 

 found in all cases. 



F 2 



