AND L]NES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 211 
I f / p2 /i , D 2 /i 9% , D !/j 9&i\ 
p 2 \Ite 2 DarDtq 3x D&DJj 0,/7 
, „ D/i D/. 
+ R 2 Da Da 
_i_ - OK , py, ^ , _P%_9^\ 
^ 1 \D.b D.rDa, 0* D.rD& 2 0se/ 
. . . ( 121 ). 
Substituting the values of dctjdx, dbjdx from (52), there is in d' : A/dx 2 the term 
^f-2 
[x, x] \x, a{\ \x, 5J 
Ov«i] [«n «i] K- 
l>> & i] Oi> M [&i, &J 
[«n «i] [«n &i] 
[ c h> 9J [9 l5 5]] 
which requires examination when x = £, y = 7), z = £, the coordinates of a point on 
the conic node locus. 
Now in this case ct v b l are roots of D/j/Dcq = 0, I)f 1 /Db l = 0. 
Hence 
[a, £] (30 + [a, V ] (St;) + [a, £] (80 + [a, a] (8a) + [a, (8/8) 
4* terms of the second and higher orders in Sf, St;, S£, Sa, S/3 = 0 . (122) 
[A £] (§0 + [/3, 7}] (St;) + [/3, Q (SO + [A a] (Sa) + [/3, /3] (S/3) 
+ terms of the second and higher orders in S^, St;, S£, Sa, S/3 = 0 . (123). 
Multiply (122) by [a, /3,], (123) by [a, a] and subtract, the terms of the first order 
in Sa, S/3 disappear, and the equation obtained is of the form :— 
(terms of the first order in S£, St;, S£) 
+ (terms of the second and higher orders in S^, St/, S£, Sa, S/3) = 0. 
Hence if S^, St;, S£ are of the order of the infinitely small quantity e; then Sa, S/3 
are of the order of eh 
Hence the principal terms in (122) and (12.3) are [a, a] (Sa) -f- fa, /3] S/3 and 
[/3, aj (Sa) -f- [/3, /3J (S/3) respectively. 
Moreover by (122) and (123), although Sa, 8/3 are of the order e% yet 
[a, a] Sa -f [«, /3] S/3 being ultimately equal to — {[a, £] (S£) + [«, 1 ;] St; fi- [a, £] S£] 
is of the order e. 
Similarly [/3, a] Sa -j- [/3, /3] S/3 is of the order e, 
