to a Symmetrical Determinant. 351 



power of the original determinant A and is therefore a constant 

 multiple of TJ in ~ l) \ 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 \ 1} X 2 , ... X^ can be so chosen that 

 V contains a squared linear factor. Suppose for instance that 

 v n = x 2 (f>, (/> being of the degree n — 3. Then we may take v 12 = xy<f>, 

 v ls = xz(f>, v w = y 2 (f>, ^ = yz<f>, v 33 = z 2 (f> and also 



v lr = x% r , v, r = yx,, v 3r = zxr (r = 4, 5, . . . n). 

 For v lr is an (n — 1) 1C through the n points in which x = meets U 

 and hence v lr must contain x as a factor ; similarly v 2r contains y 

 and v 3r contains z. The other intersections with U are the same 

 for v lr , v 2r , 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 Z7=0 in n distinct ordinary points, 

 c lf c 2 , ... c n . Then we may take v a to touch U at c 2 , c 3 , ... c n , v 22 at 

 c 1} c 3 ... c n , v rr at c lt ... c,._!, 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 — the quadratic expression becomes a perfect 

 square and the condition is linear. 



Since now v n v 22 — v l2 = Uw U22 



v 12 = passes through c u c 2) ... c n and must contain 2 as a factor ; 

 the same is true for v 13 , v u — Hence the only terms in A that 

 do not contain z are those in the leading term VnV^ ••• 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 u , 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 

 Xi, \ 2 , ... \ n , 0. 



