PROFESSOR A. CAYLEY OR THE SINGLE 
920 
which new expression we may then substitute in the equations first obtained : we 
thus arrive-at the six equations 
u+u' u—u' u+u' u—u' u + u'u—u' u + u’ u—uJ 
C . A — A . C D.B - B . T) Dvf.Bu' 
1) . L + B.D C.A + A.C — Gu'.Av'’ 
_ B . A - A . B _ D , C - C . I) Du'.Ou’ 
D . C + C . D B . A + A . B Bu'.Au 1 ’ 
B.C — C.B IJ.A — A . D Du'.Au' 
D.A + A.D - B.C + C . B Bu'.Cn'’ 
where observe that the expressions all vanish for u— 0. 
44. Taking herein u' indefinitely small we obtain 
Au.C'u — CuA'u 
BuJYu-Du.B'u 
D'O.BO 
-Iv 
Bu.Thi 
Cm.Am 
— C'O.AO -- 
Au^B'u —BuA!u 
Cm.D'w—D if.C'w 
D'O.CO 
-Iv 
GuJdu 
Qv.Bu 
“ AO.BO — 
Cu.B'u-Bu.C'u 
Aw.D'w—D? cA'm 
iro.Ao 
-Iv 
Aw. Dm 
B n C u 
" B0.C0 _ 
where the last column is added in order to introduce K in place of D'O. 
45. These formulae in effect give the derivatives of the quotient-functions in terms of 
quotient-functions : for instance, one of the equations is 
d D?t ^Bu C u . 
du A u Ait Au 
substituting herein for the quotient-fractions their values in terms of x, this becomes 
a/—= -K a/H v /l ~ x - c - x . =-K a/- ' /l '- xx ~ x 
V a—x V S® a — x V a 
A 
du 
a a —x 
_if 
2 1 
da 
or the left hand beino- = , _— 
° {a—xy^/d—x du 
, this is 
K di 
f \/af .dx 
a / a—x.b—x.c—x.d—x 
where on the right hand side it would be better to write ^/ — af in the numerator, and 
x — d in place of d—x in the denominator. 
