AND LINES IN THE INTERSECTIONS OE A SYSTEM OF SURFACES. 
151 
_ , 3 3 Q 3Q /I)/i D/, 3*A 3_Q D/, Q ( Dy t D 2 /, 3a,\ , , 
3^03/ ^ 0a;0?/ 0c \D*/ Dcq 3y/ 3y lAc 0 \D.cD_y D^Dcq 3 y) ' 
To find 3 ajdx, dajdy it is necessary to use the equation 
D f(x, y, z, «i) _ Q 
This gives 
_»5L + 5A^_ 0 (44) 
DiDj, t Do,* . h 
= 0 . (45) . 
D// Dcq Dfq 2 3y ' ' 
Now reserving the case, according to the remarks in the Abstract (‘ Proc. Roy. 
Soc., vol. 50, p. 180) and Art. 1 , in which 
b|= 0 .( 46 >- 
for further consideration, because, in this case, dajdx, 3cq/3y, both become infinite or 
indeterminate, it follows that 
^ = . U7) 
dx D/c DaJ Dtq 2 ' ’ 
s Jh = _ _d%_ m (48) 
dy Dy Do,/ Do, ! .' 
Hence 
3JA _ 3/Q , 3Q 5A 0 {Dy, D% _ / py, 71 /EA U9) 
da? dx 3 Jl T 3o! Do; T Do, ! ^ Da,/ J / Do, s . K >’ 
= s/Q f , 3QM , 3Qw, , 0 f py, d% _ py py, 1 /py ,. n) 
dxdy dxdy^ 1 dx Dy 3 y Dx \DxDy Dcq 2 Da; Dcq Dy DcqJ / Dcq 2 ' 
Hence, if rj, £ be a unode on the surface ( 10 ), and cq become equal to a when 
x = £, y = rj, 2 = £, then by means of (38), 
3 2 A 3 2 a 
— 0 -= 0 
3H ’ 3x dy ’ 
when x — y = y, 2 = £. 
Similarly all the other second differential coefficients of A with regard to x, y, z 
vanish when x = £, y = rj, z — £. 
Hence, if U = 0 be the equation of the surface locus of unodal lines, A contains U 3 
as a factor. 
