to a Symmetrical Determinant. 
351 
power of the original determinant A and is therefore a constant 
multiple of JJ (n ~ 1)2 . Taking the factor U n ~ 2 out of each row we 
have the result that the determinant of the linear expressions /3 is 
a constant multiple of U, so that the desired reduction has been 
accomplished, unless the determinant A of the expressions v is 
identically zero. 
Now this will be the case if \ u \. 2 , ... can be so chosen that 
V contains a squared linear factor. Suppose for instance that 
v n = afcj), (f> being of the degree n — 3. Then we may take v 12 = xycf>, 
v V3 = xztfy, v 22 = y 2 <j>, ^23 = y Z( f>, v 33 = z 2 (j> and also 
Vir = XXr, V. 2r = yxr, % r = (r = 4, 5, . . . n). 
For v lr is an ( n — l) ic through the n points in which x = 0 meets U 
and hence v lr must contain x as a factor ; similarly v 2r contains y 
and v 3r contains The other intersections with U are the same 
for v lr , Vw, v 3r . Hence the second (n — 2) ic factor is the same for 
each. 
The first three rows of A are now the same but for the factors 
x, y, z\ hence A vanishes with all its first minors and the method 
fails. 
Suppose on the other hand that no curve of the system V 
breaks up into an (n — 3) ic and a double straight line. Take any 
line, say z — 0, cutting U — 0 in n distinct ordinary points, 
Ci, c 2 , ... c n . Then we may take to touch U at c 2 , c 3 , ... c n ,v 2 2 at 
Ci, c 3 ... c n , v rr at Ci, ... c r ! , c r+1 ... c n . The curves of the system V 
thus determined are all different, since otherwise a curve of the 
system would consist in part of the line z — 0, which is against our 
supposition. It may be noticed also that the curves v n , v 22 ... are 
uniquely determined, for the condition that V should pass through 
a given point is generally a quadratic in X 1} X 2 . . . , but when the 
point lies on U = 0 the quadratic expression becomes a perfect 
square and the condition is linear. 
Since now v n v 22 — v 12 2 = Uw U22 
v 12 = 0 passes through c u c 2 , ... c n and must contain z as a factor; 
the same is true for v l3 , v u Hence the only terms in A that 
do not contain are those in the leading term v n v.> 2 . . . v nn which 
certainly exist, and therefore A does not vanish identically. 
There is therefore one reduction of the ternary quantic to the 
form of a symmetrical determinant with linear constituents for 
every theta-function of even characteristic which does not vanish 
for zero values of the arguments. (Compare Baker, Abelian 
Functions, pp. 268 — 270.) 
When U has been thus reduced the functions v n , v 12 ... are the 
first minors of the determinant and V may be derived by bordering 
U with a row and column each consisting of the quantities 
A-i, \ 2 , ... \ n) 0. 
