8 MR. E. CUNNINGHAM ON THE NORMAL SERIES 



Now the way in which the successive equations follow one another shows that the 

 coefficient of aV in 6* is equal to that of a:, 1 in ft+r _,. 



Thus 0',,-r+i being independent of a^ 1 , l p - r is independent of x 2 \ and in general, 

 l p -,+ k (k = l...r-l) being all independent of a^ 1 , Q l p - r does riot contain x^..^,^. 



Thus, if the assumptions made on p. 7 are satisfied for any particular value of r 

 less than p, they are satisfied for a value of ? one greater than that value. 



For ? = 1 the statements have been justified, and it follows therefore that fl 1 ^...^ 1 

 are all determined uniquely without the knowledge of a;, 1 , x 2 \... from the first (p+l) 

 of the equations X, and by the same equations x^-.x^ are found, except for their first 

 elements, the expressions obtained containing those first elements. 



5. Consider now the (p + 2) Ul equation X in regard to its first element. 



As before, this will be independent of x p l ...x 3 l ; but on account of the extra term 

 arising from dy/dt, which now enters for the first time, the coefficient of aV is not 

 zero. 'It is, in fact, 1. 



Thus, the quantities 6 l l ...6 l p+l being now known, this equation gives a^ 1 . 



Similarly, the next group's first member will contain the term ZxJ but will not 

 contain x-}.,,x l p + lt and will therefore give x 2 l after o^ 1 is found. 



Thus all the elements x^ are determined successively, and returning to the 

 expressions for a?/ (?'>l) in terms of these and substituting the values so found, all 

 these are given also. 



The equations for the columns y, z, &c., being treated in the same way, give the 

 corresponding O's uniquely, and also the coefficients in the series of which these 

 columns are composed. 



Thus it is shown that when the "characteristic equation" p+ i p\=0 has its 

 roots all different, the equation 



dy/dt = ny, 

 where 



a p+ i being in its canonical form, possesses a unique formal solution in the form 



where the elements of Xp -- Xi llot "i the diagonal are zero, and the elements of r? are 

 power series in t, reducing for t = to the matrix unity. 



The matrix n^ + ... + j can at once be written in the form a>/a> , where o> is 

 a matrix whose non-diagonal elements are zero, and whose k ih diagonal element is 



e p- tr 

 and w is the value of CD at t t lt . 



