AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
169 
[6 f) (§0 + [£ 0 (SO + [0 Q (so = o, 
bi> 0 ) + bi> 0 (SO + bi, 0 (SO = o, 
ft « ( s O + ft 0 (§0 + ft Q (SO = o. 
These equations must be consistent, and therefore 
ptcmo. mi _ ( 
d[ ?, v , n ~ 
This is the condition that the tangent cone at 0 77 , 0 viz. :— 
k f] (x - f r + [,.,] (y - ,)* + [c g (z - if 
+ 2 [,,Q(Y-,) (Z-£) + 2 [{, f] (Z-0(X-f) + 2 [f. 7 ](X-f)(Y-,) = 0 , 
may break up into two planes. 
PART II.—THE EQUATION OF THE SYSTEM OF SURFACES IS A RATIONAL INTEGRAL 
FUNCTION OF THE COORDINATES AND TWO ARBITRARY PARAMETERS. 
Section I. (Art. 1). —Preliminary Theorems. 
Art. 1 . (A.) If 0 7 ], £ are the coordinates of any 'point on the locus <p (x, y, z) = 0 
(where <f> is a rational integral indecomposable function of x, y, z), and if the 
substitutions x = £, y — r\, z — £ make \p (x, y, z) and all its partial differential 
coefficients with regard to x, y, z up to the n 1h order vanish, and if they also make 
any one of the partial differential coefficients of the (n + l) ih order vanish, they 
will also make cdl the partial differential coefficients of the (n + \) <h order vanish 
(ip being a rational integral function of x, y, z, but not in general indecom¬ 
posable). 
Suppose that when x = £ y = y, z = 0 
d n + 1 \p' 
dx r +1 dy s 0 ,j" - r - j 
where 3 denotes partial differentiation when x, y, z are independent variables. 
To prove that the same substitutions make 
and 
8 ,i+1 ^ _ 0 
dx r dy s ^ 1 dz n - r ~ s * 
3 ,i + 1 -Jr 
-r-— a 
dxrdy s dz n - r ~ s+1 
MDCCCXCII.—A. 
Z 
