DIFFERENTIAL EQUATIONS OF THE n"' OrDER. 



39 



Help + (K+VG)dq ^ AliLv = . . . . (76) . 



Now (74), joined with either (75) or (76), constitutes the usual 

 auxiliary system of the equation of Ampère. 



If iV =: , (74) and (76), of course, reproduce the Mongean 

 auxiliary system. 



3) The equation of the third order: 



Az,^ + ß~2, 1 + C-^i. 2 + 144. 1 — ^-3, ^1, 2) — C/ = ... (77) 

 gives 



{A — L:, ,) m' —{B + 2L:,, ,) m' + (<^ — L:,^ «) m = 0, 



whence m^ = and m^ and m^ are given from 



B + 2Z-2,i + y G 



putting 



m = 



G = 



(78), 



2{A — Lz,,,) 



B^ — AAC -[- ALU. 



From (78) we get 



J. _ ZJ + 2Lc,,, + K^ 



m - 2(C-Zr3,o) ^"'^' 



Now the equations (78) and (79) easily give the auxiliary equations 



B ±yG 



Ady — Ld:\ j 



dûs, 



C die — XtZcg, u = 5 — *^y 



(80). 



This system is incomplete, because tlie root m = has been 

 omitted. That it also only gives particular integrals, it may be enough 

 to shew in a special case. Let, namely, B and C be both 0; then the 

 equation (77) becomes simply 



^.3,0 + i^G1,l---3,0^-.,2)— t/= .... (81), 



and its auxiliary system is 



Ady — Ld:,, = ±yLZJ.dœ (82), 



Ldz.^o = + yLÜ.dy 



(83), 



and, supposing the coefficients to be constant, we get from these equations 



