328 
Proceedings of Royal Society of Edinburgh . [sess. 
The Bliminant of a Set of Quaternary Quadrics. By 
Thomas Muir, LL.D. 
(Head December 7, 1896.) 
1. The “dialytic” method of elimination, in the case of more 
than one variable, is not without its drawbacks, as most mathe- 
maticians know. The requisite derived equations are not always 
easily obtained, the difficulty being due as often to the existence of 
too many as of too few ; and, when this has been got over, it not 
unfrequently happens that the order of the resulting determinant 
is alarmingly high, and unaccompanied by any hope of a successful 
guess as to the character of the extraneous factor. The discoverer’s 
original paper * affords sufficient testimony of this, and very little 
has been done since its appearance to put matters on a sounder 
footing. The most noteworthy improvement, due to Cayley,! is 
more interesting theoretically than practically, his main object 
beino- the detection of the extraneous factor when there is an over- 
o 
plus of equations. Nothing, indeed, will be found more conducive 
to an understanding of the limitations of the method than a careful 
comparison of the application of this process of Cayley’s to the 
problem of eliminating x , y, z from the three ternary quadrics 
a^x 1 + bgf + cf 2 + Igjz + m Y zx + nyxy = 0 j 
a.;x 2 + b. 2 y 2 + c. 2 z 2 + Ig/z + myzx + n 2 xy = 0 y 
a.f 2 + b./y 2 + cf 2 + lyjz + myzx + n.gxy — 0 j 
with Sylvester’s original treatment of the same problem, and the 
latter especially as commented on in footnotes by the author 
himself. 
2. The want of definiteness in the mode of arriving at the exact 
* Sylvester, “Examples of the Dialytic Method of Elimination as applied 
to Ternary Systems of Equations,” Camb. Math. Journ., ii. (1841), pp. 
232-236. 
t Cayley, “ On the Theory of Elimination,” Camb. and Dub. Math. Journ., 
iii. (1848), pp. 116-120; or Collected Math. Papers, i. pp. 370-374. 
