DERIVATIVES ASSOCIATED WITH LINEAR DIFFKKKNTIAL EQUATIONS. 4G7 



aa* + 26ay + cy 2 = XA "I a? a* + 26'ay + c'y 2 = X A' 1 



aa -f 6 (J3y + aS) + cyS = XB I , a'aft + 26'()8y + a8) + c'yS = XB- L 

 a* + 2b/38 + cS 2 = XC J a'0 2 + 26'08 + c'S 2 = Xtf J 



six equations, apparently, and really five equations involving the ratios of the four 

 quantities a, ft, y, 8, so that two conditions must be satisfied among the constants of 

 equation (50). We at once find 



X 2 (AC - B z ) = (aS - y) 2 (ac - 6 s ), 

 X s (A'C - B' s ) = (8 - y) 2 (aV - &"), 

 X 2 (AC' + A'C - 2BB') = (a8 - j8y) 2 (ac' -f a'c - 266'), 



and therefore the two necessary conditions are 



AC-BE AC 7 4- A'C - 2BB' A'C' -B' 



ac - V at + a'c - 2W a'<f - i" ' 



Assuming these conditions to be satisfied and denoting the common value of the three 

 functions by I* 2 , we have 



ecS - /8y = XR 



To find the value of the ratios a : ft : y : 8 we write 



y = a6, 8 = fa a = #/,, 



so that 6, <f>, \ji are the quantities to be determined. The first of them can at once be 

 obtained from 



a + 2M + effi A 



and the second from 



a + 26ft + ctf C 



a' + 2V$~+c'<t> 3 '' ' C' ' 



From the first three equations we have for any value of 



(+ W + 26(a+68)(y + 8) + c ( r + S) 2 : 



It follows that 



cXP 2 = Atf 2 - 2Ba/3 + Ca 2 ; 



and similarly from the second three that, 



c'XP 2 = A'0 9 - 2B' + C'a 8 . 



[It may be remarked that these are the types of the equations which would have 

 been obtained if substituting for x in terms of s from (51) had taken place in (50)]. 



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