Prof. Newman on Determinants. 891 
Pee Ol Boe ea) wee 6. vl Oem Of 8) ake me 
Bv+ B,v, + B,v,+.. a 
: Nv+N,v,+N,v,+...+N,=0 
and the solutions are denoted by » 
mo=a,; m'v,=a,; ...m'v,_,=On_)3 
we get the relations 
m=Aa,+A,a,+A,a,+...+A,_,4, ,+A,m! ; 
a= Pa, + P.a,+Pja,+...+P,_, aaa 
out of which flow all the rules for the genesis of Eliminants, and the 
application of them to solve linear eqq. of any degree. 
In adapting the theory to the proof of elementary propositions, as, 
in forming the Product of two Eliminants, the paper urged the uti- 
lity of the principle, that every Eliminant is a linear function of any 
one of its columns, and also, of any one of its rows ;—which prin- 
ciple may often be so applied as to show by inspection, a priori, that 
certain constituents are excluded from this and that function, and 
thus enable us to obtain its value by assuming arbitrary values for 
such constituents. It deprecated (at least for elementary uses) the 
notations used by Mr. Spottiswoode* and others, not only as invol- 
ving needless novelty to learners, but because no page can be broad 
enough to afford to write 
(1, 2)(1, 1)'+(2, 2)(1, 2)'+(3, 2)(1, 3)! instead of BX+6Y + Z, 
and because accents, so related, are hard to see in a full page, and 
the general aspect of every element is so like that of every other ele- 
ment, that the fatigue of reading soon becomes confusing and in- 
tolerable. 
2. But the main topic of the paper was to advocate the use of 
Eliminants in Geometry of three dimensions, especially in every 
systematic treatise on Surfaces of the Second Degree. Various illus- 
trations and results were given, which the writer believed to be new ; 
on which account, some of them may be briefly noticed here. 
Problem. “To find the length of a perpendicular p, dropt from a 
given point (a4 c), on to a given plane /x+my+nz+p=0; when 
the axes are oblique, and the cosines of the angles («y)(xz)(yz) are 
gwen; viz.=D, Ki, F.” 
Result. Take G and H to represent the eliminants 
GoM Di Fl and EN shat 
EF 1 vs 
~mn 0 
then p is known from the eq. 
pVH=(la+mb+ne+p) VG. 
When ¢ is given, this eq. determines the relations between / mn p, 
* It may be right to state, that Mr. Newman opened the paper by a grateful and 
honourable recognition of Mr. Spottiswoode’s labours. 
