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Proceedings of the Royal Society 
branches thereof, the order of the curve, its singularities, class, 
&c. ; also in regard to the v- zomal curve Jl(® + + &c. = 0, 
where the zomal curves 0 + L <3> = 0, all pass through the points of 
intersection of the same two curves © = 0, = 0 of the orders 
r and r — s respectively ; included herein we have the theory of the 
depression of order as arising from the ideal factor or factors of a 
branch or branches. A general theorem is given of “ the decom- 
position of a tetrazomal curve,” viz., taking the equation to be 
Jiu + + JJT = 0 ; then if U, V, W, T 
are in involution, that is, connected by an identical equation 
a?7-pbF+cTF'+d2 7 =0, and if l , m, n , p, satisfy the con- 
, ... I m n p 
Clltlon aT’*'!) **■ *c" + ^ the tetrazomal curve breaks up 
into two trizomal curves, each expressible by means of any 
three of the four functions U, V, W, T ; for example in the form 
s/V U + Jm' V + Jp' T — 0. If, in this theorem, we take^> = 0, 
then the original curve is the trizomal flU + JmV + fnW— 0, 
T is any function = -^-(a^7+bF+c W), where, considering 
l , m, n as given, a, b, c are quantities subject only to the condition 
l m n 
— + y- + — = 0, and we have the theorem of “ the variable 
a d c 
zomal of a trizomal curve,” viz., the equation of the trizomal 
JIU + JmV + JnW = 0, may be expressed by means of any 
two of the three functions U, V, W, and of a function T determined 
as above, for example in the form VI' U + Jm' V + fp'T = 0; 
whence also it may be expressed in terms of three new functions 
l 7 , determined as above. This theorem, which occupies a promi- 
nent position in the whole theory, was suggested to me by Mr 
Casey’s theorem, presently referred to, for the construction of a 
bicircular quartic as the envelope of a variable circle. 
In the v-zomal curve ^(©-i-Z/b) + &c.= 0, if © = 0 be a conic, 
$=0a line, the zomals © + Z4> = 0, &c., are conics passing through 
the same two points 0 = 0, <3> = 0, and there is no real loss of genera- 
lity in taking these to be the circular points at infinity — -that is, 
in taking the conics to be circles. Doing this, and using a special 
notation A° = 0 for the equation of a circle having its centre at a 
given point A, and similarly A = 0 for the equation of an evanes- 
