A Treatise On Linear Differential Equations

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: (16) p, = - {a, + a, -{- . . . -{- a„), A = 2a, ay-{n-i) -j^ - {n-2) -^ - ... - dx dx dx ' A = — ^ ^i'^j^k— \n—i){n—2)d''a, {n — 2){n — ^)d''a.
I . 2 dx' f I . 2 dx -4-..
+ («-2)(«, +^3+ . . • + ^«) 5^ +('^-3X^3+ . . • +^«)^ + + «: dx dx + ^.
+ DECOMPOSITION INTO SYMBOLIC PRIME FACTORS. 393 These formulae give the coefficients/ in terms of the coefficients ^, and consequently, since the as are functions of z^, , z', , . . . , z/„, in terms oi v^, v^, . . . ^ v^. Write, for brevity, (17) Pi =f
...i{(i, a^, ' • ', ««); now, since p^, p^, . . . , p„ have the same values whatever be the chosen system of fundamental integrals of the differential equation, and consequently whatever be the chosen system of the functions V, it follows that the functions /"are invariants. \i a^, a^, . . . , a^ are constants, we have (18) /, = — 2^, -, /, = ^Uittj, A = — '^aiafik, • • • » and therefore /j, A» • • • ^""^ the coefficients of the algebraic equa- tion {t — aXt - a, ) .

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