662 
PEOFESSOE SYLVESTER. ON THE EEAL 
from which we may easily deduce 
|=2 (I)- (|)M(|) 
in which it will be observed that the indices of the powers of v are precisely comple- 
mentary to those in the preceding expansion, the two series of indices together com- 
prising all multiples of ^ from positive to negative infinity. 
(96) We now see how, supposing the curve to be given with £ and ri at any angle, 
K. L 
the axes corresponding to — , — may be defined : viz., the origin of coordinates will be at 
J J 
the cusp-node ; q, along which - is reckoned, will be in the direction of the tangent at 
that point ; and |, along which — is reckoned, will be the axis of that common parabola 
which at the same point has the closest contact with the given curve. 
It seems desirable, with a view to a more complete comprehension of the form of the 
amphigenous surface, i. e. the limiting surface of invariantive parameters, to ascertain 
the nature of the systems of plane sections of it, parallel to each of the three coordi- 
nate planes. The sections parallel to J, which are curves of the fourth order, have 
been already satisfactorily elucidated. It remains to consider briefly the sections parallel 
to J and D, which will be curves of the ninth order. 
(97) When L is constant, writing J=z, D=y, where for facility of reference we may 
A 3 
conceive y horizontal and 2 vertical, and making we have 
z 3 =ky(<p- i-l), 
MH)_p 
(p-3)(p + 2)» 
( 1 +# 
8y_2 fo-l)(4f> + 3) v 
8z 1 4p + 3 v 
7-3 
8* 1 (f + 1)* 
y~3 (p + 2)(p-3)(<p + l) 0 ^ 
ly~2k (f — l)(f + : 
when tp — — 1 , 
* = 0 , 
y— qo, 
„ P=-t, 
0 
II 
Qyf 
O 
„ <P= 0 , 
* - 0, 
1 — 1 
1 
5m 
„ <P= 1 > 
0 
II 
„ <p=H-co , 
z = + 00 , 
5* ^ 
II 
+ 
8 
„ <P=- 2, 
y = 0 , 
8 y~ 00 ’ 
„ <p=— CO, 
S =+CO, 
y =+00 . 
Hence it appears that the curve consists of three branches, two coming together at 
an ordinary cusp at the point corresponding to p=— f, and the third completely sepa- 
rate. The nature of the sign of k does not affect the nature of the curve. If, for 
greater clearness, k be supposed positive, the first branch, having the negative part of 
