AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 227 
Now the only relation between §£, Sr), S£ is that which expresses that the point 
g -f- S£, r) -f- Sr), £ + S(, is on the tangent plane to the locus of intersections at £, 17 , £. 
Hence 
0P /HQ_HP /0Q_0P /0Q 
9£/ S% ~ S v / Srj ~ S£l S£ 
It is now possible to determine the values of a, b which are indeterminate as given 
by (145). 
For, representing the value of a in (145) by the equation 
a 
P/Q, 
it follows that the true value of a is the limit to which the expression 
Q + l s f+a>+a> 
approaches, when Sr), S£ vanish. 
Now P = 0 , Q = 0 ; hence by (159) the true value of a is equal to any one of the 
three ratios in (159). 
Besides the values of a, b given in (145), (156), other forms may be obtained from 
equations (143), (155). 
Putting these together 
WU - ?;V _ mo - U 2 UY - Ww 
— uv _ W 2 — WU -Vv~ WV - U u [ 
WY - uU UV - Ww uw - V 2 ( 
— uv - W 2 — WU — Yv WV - U u j 
(160). 
All these values are indeterminate. 
Now although the value of each of these fractions can be found by differentiating 
numerator and denominator with regard to any the same variable, yet they will not 
all lead to the true value of a, b, because the true values of a, b are found by 
solving the equations ua + W b + V = 0 , W« + vb U = 0 , and finding what the 
values approach to as the coordinates approximate to the coordinates of a point on 
the locus of ultimate intersections. Now at points not on the locus of ultimate inter¬ 
sections, the values of a, b do not satisfy Ya + U6 + w — 0. Hence the true values 
of a, b cannot in general be found by solving this last and either of the preceding 
equations, and then finding the values to which these approach as the coordinates 
approximate to the coordinates of a point on the locus of ultimate intersections. 
The true values are obtainable only from the solutions (145). 
2 G 2 
