192 Sir James Cockle on the 



Consequently when (1), (2), (4), and (5) are satisfied identi- 

 cally, and tj is determined from (3), and/, g, h, and k from the 

 outstanding equations, the factors are in arithmetical pro- 

 gression. 



15. Let n=3, then 



XW = - 2L "+ (M - 8L> 3 + (4N - 5M> 6 - 2N^ ; 



and if the terordinal in z is of the form 



../// 



+ 3s2 / + |/^=0, 



the conditions (4) and (5) are satisfied identically. Moreover 

 2B = 95 identically; and if 6=f, (1) is satisfied identically, as 

 is (2). But this last terordinal can, if 46 = 9 be linked with 



a biordinal (compare Proc. L. M. S. vol. xiii. p. 65, art. 41, 

 p. 67, art. 47, p. 68, art. 52, and p. 70, art. 61). 



16. Next, if we start from 



we can derive from it 



y'" + Z\y" + Set/ + (3/ + 9Xs)y = 0, 

 whence, taking the criticoid, 



^ + 3( s -X 2 -Vy + (3/ + 6\s + 2X 3 -V / >=0. 



17. Write this in the form 

 z f " + 3s 2 z' + t 2 z=0, 



and define X by 

 we get 



3X£>X=A + Ea' 3 : 



XV«, = L 2 + M 2 ^ 3 + N 2 # 6 , 

 XV* S = P 2 + Q 2 .* 3 + E 2 ^ 6 + S 2 A' 9 ; 



and we now deal with the suffixed quantities (s 2 , L 2 , . . ) as 

 we did with those unsuffixed (s, L, ..), but ultimately ex- 

 pressing the suffixed in terms of the unsuffixed quantities. 

 Thus Ave take the system (w = 3), 



B 2 = i(9 + V / 9^4^)6 2 , (1) 2 



C 2 + 6 2 =-K N 2-L 2 )(3 + V / 9^46 2 ), . . (2) 2 



^=-1(3 + ^9^4^2) (3) 2 



18. Developing, we find 



L 2 = L-A 2 + A, 



M 2 = M-2AE + 4A-2E, 



N 2 =N-E 2 + E: 



