120 Proceedings of the Royal Irish Academy. 



Let the equations be 



^11 (!));. + ^,3 (l»)y = 0, (22) 



^,,(/;),r+.^,,(i))y-=0, (23) 



of orders m in x and n in y, i.e. the highest derivati%^es being D'" (x) and Z)" (y). 



The characteristic determinant, with D as the " variable," is 



which will be denoted by A (Z)). 



(24) 



§ 3'1. TJie solvMon, for certain initial data, stated and verified. 



When the determinant is " normal " we may obtain a solution which 

 satisfies the conditions that initially x and its derivatives up to the [rn - l)'*, 

 y and its derivatives up to the {n - 1)"', inclusive, shall have arbitrarily 

 assigned values. 



The solution in x is 



2Tfix 



%\ 



e^-'dX 

 "MA) 



^ + ^: ^ 2/) <pi".{^) 



D -X 



^2> jD) - (l>n (A) 



U - \ 



<P%1 [B) - ^32 (A) 



y, ,j,,i{\) 



(25) 



1 = 



1)-X D -X 



the determinant in this is obtained from A by altering, as indicated, the 

 constituents of the first column. For y, the elements of the second column 

 of A, instead of those of the first, are replaced by those of the first column 

 in (25). 



In the very special case of m = 0, x,^ does not occur in the solution ; each 

 numerator i)i the first column of the determinant has the form c - c, where c 

 is constant. 



As the symbolic notation is somewhat puzzling, I give in the more usual 

 notation the value of x for the equations 



(«„i)' + huD + Cn) X + [a,.D-- + hnD + c,^) y = 0, (26) 



(«„D'- + h.J) + c.a)x + {a,,D' + h^D + c,,) y = 0. (27) 



It is 



2ir)x = 



a 



fdX.e>^t 



A (A) 



(«„A + &„).f„ + iiuXi + («,2X + hn)y„ + «,22/i, flijA' + b^X + c^ 



((7j,A + SzO-I'd + "21^1 + (flasA + ^22)^0 + *222/l, ffjjA" + 622A + C21 



(28) 



2^01 ^u !/o> Vu being the initial values of .r, dcc/dt, y, dy/df. 



lleturning to (25) and its analogue for y, in the first place these 

 expressions satisfy (22), (23). In fact, these differential equations are satisfied 

 by values of the forms of (25) and its analogue if the constituents of the 

 first column in (25) are replaced by anj^ analytic functions, 1^1 (A), -ip^ (A), 



