ON POLYZOMAL CURVES. 
490 
[414 
viz., the trizomal curve \/lU + vmV + *JnW =0,—if a, b, c be any quantities connected 
by the equation 
l m n 
- + - + - = 
a b c 
(the ratios a, b, c thus involving a single arbitrary parameter); and if we take T a 
function such that aU + bF + cW + dT = 0; that is, T= 0, any one of the series of 
curves aZ7+bF+cTr=0, in involution with the given curves TJ = 0, F=0, W = 0,— 
has its equation expressible in the form 
a >JmU-b *JlV+ nT = 0 ; 
that is, we have the curve T = 0 (the equation whereof contains a variable parameter) 
as a zomal of the given trizomal curve VZt7 + VmF+Vn]F=0; and we have thus 
from the theorem of the decomposition of a tetrazomal deduced the theorem of the 
variable zomal of a trizomal. The analytical investigation is somewhat simplified by 
assuming p = 0 ab initio, and it may be as well to repeat it in this form. 
47. Starting, then, with the trizomal curve 
l '/lU+ VtoF + \fnW = 0, 
aU + bF+ cTT+dr=0 
and writing 
as the definition of T, the coefficients being connected by 
l 
— + 
a 
m 
+ 
= 0, 
the equation gives 
IU+ mV + 2\/lmUV-nW = 0; 
or substituting in this equation for W its value in terms of U, V, T, we have 
(an + cl) U+ (bn + cm) V + 2c Vfon UV + dnT = 0, 
which by the given relation between a, b, c, is converted into 
- ^ mU - — IV + 2c \/lmUV+ dnT = 0; 
b a 
that is 
a 2 mU+hHV- 2ab y/lmUV = — nT, 
c 
viz., this is 
(a ^mU- b VFF) 2 = -^ d nT, 
or finally 
aVmF-bVTF+y/— nT= 0.