164 PROFESSOR CHRYSTAL ON THE EQUIVALENCE OF 



where D stands for cl/dt, and f v f 2 , etc., are integral functions of D with constant 

 coefficients. Then the conclusion is drawn from (1) that 



Ke = 0, Ky = Q, Kz=0 (2), 



where K is the determinant \f (D), g 2 (D), h s (D) | ; so that each of the variables satisfies 

 the differential equation 



K£=0 (3), 



which we call the " characteristic equation " of the system, K being the " characteristic 

 determinant." The system (2), however, is not equivalent to (1). Hence, if 



£=A X e K ^+ . . . .+A n c K nt 



be the solution of (3), (n being the order of that equation, and therefore the degree in D 

 of the integral function K), it does not follow that 



n n n 



x = X A„e A »* , y = X B n e knt , z = 2 C w e A »i', 

 ill 



containing dn arbitrary constants in all, is the solution of (2). To get rid of the 

 superfluous constants the values of x, y, z are substituted in (1), and we thus get a set of 

 equations of the form 



/, (\) \+ ffl (X x ) B^ (X x ) C 1 = , 



/ 2 (X 1 )A 1+5 r 2 (\ 1 )B 1 +7 i2 (X 1 )C 1 = 0V .... (4). 



/ 3 (X,) A, +g 3 (X x ) B. + h, (\) C 1 = ) 



&c. 



Since X is a root of the equation K A = 0, obtained by substituting X for D in K, the 

 equations (4) are equivalent to any two of them. If these two be independent, and 

 neither of them identical as regards A l5 B x> C 1? the ratios of A 1? B l5 C x are determined ; 

 and, on the like assumptions for X 2 , . . . ., X n , the 3n arbitrary constants reduce to n. 

 Even if all the assumptions made were generally true, this process could scarcely be said 

 to be a very satisfactory proof that the order of the system is really n. In point 

 of fact, however, as is well known, the assumptions made are not always true ; and, 

 in particular cases, the equations (4) do not in fact determine the ratios of A 1} B l5 C r 

 It is true that there are various ways in which the solution may be amended in particular 

 cases, but no general process, so far as I am aware, has been given which amounts 

 to a proof of the theorem that the order of the system is always the same as the 

 order of the characteristic differential equation, i.e., the degree of the characteristic 

 determinant. In his remarkable memoir, " De investigando ordine systematis aequa- 

 tionum differentialium vulgarium cujuscunque" (Crelle's Jour. lx. 1865), Jacobi makes 

 this theorem his starting-point ; but the proof which he gives begs the question, 

 for it amounts to nothing more than what has just been sketched for the case of 

 three dependent variables. 



