The Collected Mathematical Papers of Arthur Cayley volume 4

Arthur Cayley

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But in this case K is of the same species with K, viz. , of the species I. , and by the theorem, K (X, Y) = (6* X, W F) where of = d (c + d ) -l, ij = a (a -b )-l (mod. 16) ; and we ought to have that is a/ = r, y = y (mod. 16), d (c + d )-l=d(c + d)-I+r, (mod. 16), or substituting for a, b, c, d their values a + br, b, c + dr, d, these become d(c + d + dr) = d (c + d) + r, (mod. 16), (a + br) (a b + br) = a(a b) or (mod. 16).
293] ON A CEKTAIN SYSTEM OF FUNCTIONAL SYMBOLS. 489 But r being even..., these become d 2 - 1 = 0, 2ab - b" + b*r = (mod. 8) ; or, since in the case under consideration, a, b, c, d = 1, 0, 0, 1 (mod. 2), and therefore 6 2 r = (mod. 8), the two congruences become d? - 1 = 0, a 2 - (a - 6) 2 = (mod. 8), which are satisfied since d, a, and a b are all odd. And in a similar way the theorem is proved for all the other cases.
In M. Hermite s problem, X, Y are replaced by , ^u>, AX, AY are (w + l). I/r(o) + l), and EX, BY are $( --, ^( --, the properties of the functions , ^ being such that, as in the preceding investigation, A(X, F)=0, , B(X, F) = (F, X), where 0(=e i!r ) is a sixteenth root of unity, and KX, KY are respectively equal to ( rZ"^J ^ (, ?

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