Sir James Cockle on Transformation. 359 



I find that the binomial terordinal dedncible from that in z 

 (arts. 16, 17) is 



/(A,B,I)r+/][A + 2»,B-»,J)a»C=0, . . (15) 

 from which a factor, viz. A + 2B, may be made to disappear. 



60. Getting rid of this factor, putting Z = # _B £, and a 2 

 (which is arbitrary) = 1, and changing x 3 into or, we are led to 



(D + iI)(D-P)Z + (D + 2a) + iJ)(D + 2 & >-iJ> 2 Z=0. 



61. But, the restriction entailed by the progression being- 

 removed, we can employ both the values of co given in art. 42, 

 and so obtain two equations either of which is a transforma- 

 tion of the other. 



62. In verification I add that, by a further application of 

 Boole's process, these two equations may be put uuder the 

 respective forms 



(D-/3 1 )(D-/3 2 )« + (D-« 1 )(D-« 2 >^=0, 



(D-|8 1 )(D-ft) Ms + (D + U-« 1 -2)CD + U-«,-2)^ti, = 0, 



and that one of these forms is a transformation of the other, 

 viz. u 2 =(l + x 2 ) 2 W . 



63. If in b 2 = —r) 2 (r\ 2 + 3) we for t} 2 substitute 2> — b 2 B 2 and 

 reduce, we get 



&{& + 36)(a> + l)}{& + 3(w + 2)(« + 3)} = 0; . . (16) 



and from (N 2 — L 2 ) -1 (C2 + 62) = 3— 6 2 " 1 B 2 we get 



{6 + 3a>(a> + l)}{5(ft-l) + (&> + 3)(N-L)}=0. . (17) 



64. Both (16) and (17) are satisfied if b + 3co(<o + l) = 0. 

 This is the solution which I have discussed. They are also 

 both satisfied if 6 = and o) + 3 = 0. This solution is inad- 

 missible; for it would make % a vanishing fraction. There 

 remains the system 



& + 3(ffl + 2)(a> + 3) = 0, \ (m 



&(a-l) + (Q> + 3)(N-L)=0;/ ' l ; 



which is important, but which I shall not discuss in these 

 pages. The latter solution of art. 26 is not irrelevant to the 

 system (18). 



2 Sandringham Gardens, Ealing, 

 Match 14, 1882. 



Erratum. 



Vol. XII. p. 196, art. 36, lines 10, 11, transpose " one is " and " two are " 

 and in line 12 dele integers. 



2E2 



