AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 159 
(D). It should be noticed that in the preceding section (C), the infinitesimal 
of the first order is the increment in the value of a, a root of D fix, y, z, a)/Da = 0, 
when £, y, £ receive increments S£, By, S£ respectively; and in particular that it is not 
of the same order as S£, By, S£. 
For if 3a be the increment in the value a, then 
[ a > (S£) + [a, y] (By) + [a, £] (S£) + [a, a] (3a) 
[«, £ C] (S £) 3 + K v> vl (&vY + [«, C> C] (SC ) 3 + [«, «, «] (Sa ) 2 
+ 2 
+ 2 [a, >7, £] (Sr;) (S£) + 2 [a, £, £] (S£) (S£) 4- 2 [a, £, 77 ] (S£) (St;) 
+ 2 [a, a, £] (8a) (S£) + 2 [a, a, 17] (8a) (Sr;) + 2 [a, a, £] (8a) (S£) 
+ • 
= 0 . 
Now because [a, a] = 0, this equation can be written in the form 
u l + u. 2 + 2y 1 (8a) -j- (Sa) 3 = 0, 
where the suffixes denote the order of the terms, when (S£), (By), (S£) are taken to 
be of the first order. 
Hence, if e denote an infinitely small quantity of the first order in (S£), (By), (S£), 
then 8a is of the order e 1/2 . 
And now f (x, y, z, a 2 ), when a? = £ + S£, y — y By, z = £ + 8£, and a 2 = a + 3a, 
becomes 
/(£ % C a ) + [fl (S£) + M (St;) + [£] (S£) + [a] (3a) 
[£ fl (S £) 3 + [rj, v] W + [C, £] (SC ) 3 + [«, *] (S «) 3 • 
2 + 2 Lffi C] (Sr;) (3£) + 2 [£, £] (S£) ( 8 £) + 2 [£, r;] (S|) (Sr;) 
_+ 2 [a, £] (Sa) (S£) -f 2 [a, r;] (Sa) (By) + 2 [a, £J ( 8 a) (S£) 
+ 
[I, a (S £) 3 -f l*h vl (H 2 + [C, C] (SO 3 
2 +2 [r;£] (Sr;) (S£) 4 - 2 [£, £] ($£) (S£) 4- 2 [£, >;] (S£) (By) 
„-f 2 [a, £] (Sa) (S£) + 2 [a, y ] (Sa) (Sr;) -f 2 [a, £] (Sa)( S£) 
Hence f (x, y, z, a 2 ) is of the order e s/2 when X — £ -j- 8£, y — y -j- By, z — £ + BL 
In like manner, when x — £ + S£, y = y 4 - By, z — £ 4- 3£, a x — a -f Sa', 
