166 PROFESSOR CHRYSTAL ON THE EQUIVALENCE OF 



A = | itf* Km | (8) 



the modulus of the system (6) with respect to (5). 



We shall prove that the necessary and sufficient condition that the system (6) be 

 equivalent to (5) is that the modulus of (6) with respect to (5) be constant. 



In the first place, the condition is sufficient ; for, if 



I ~i-H 2 Km | 



be the reciprocal of the determinant A, we have from (7) 



AU 1 = E 1 V 1 + H 1 V 2 + . . . + K l V m 



AU m = E m V 1 +H m V 2 + . . . +K m V w . 



Hence any solution of (6) satisfies 



AU 1 = 0, , AU M =0. 



But, if A reduce to a constant, this last system is equivalent to (5), for A cannot vanish 

 since the system (6) consists of m independent equations. The condition is also necessary ; 

 for, if the system (5) be a derivative of (6), then there must exist a set of integral 



functions of D with constant coefficients, say £/, . . . . , £„/, , «•/, . . . , «■„/, 



such that 



U 1 -£'V 1 + f 2 'V 2 + +£ m 'V m (9). 



U m = /C 1 'V 1 + K- 2 'V 2 + .... +K m 'V m . 



If we substitute the values of \J U , U wl from (9) in the identities (7), then, 



since U-,, . . . . , U m are independent, we must have 



iii-2+i-2Vo+ +£«'*., =o 



il£m + %2Vm+ • • • . + £irt'f (a = ; 

 Vi'£\ +V0V1 + • • • • +VmK 1 =0 

 Viiz +V2V0 + ■ • • • +W*2 =1 



Vi^m+VzVm + • • . . +VmKm*=0 



(fee. &c. 

 Solving the above systems we have 



£-S 1 /A,.& # -H 1 /A, , f-'-Kj/A 



K 



i' = H,„/A) /c 2 ' = H,„/A, . . . . , K m ' = K m jA 



