427 



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, l)' + (2, 2)0, 2)' + (3, 2)0, 3)' instead of BX+6Y+/3Z, 

 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 (abc), on to a given plane lx+my + nz+p=Q ; when 

 the axes are oblique, and the cosines of the angles (xy}(xz)(yz) are 

 given ; viz. = D, E, F." 



Result. Take G and H to represent the eliminants 



1 D E-l 

 D 1 F-m 

 E F \-n 

 I m n 

 then p is known from the eq. 



p VR=(la + mb + nc+p) VG. 



When p is given, this eq. determines the relations between Imnp, 

 which are the test, that the plane may touch a sphere given in 

 position. 



* It may be right to state, that Mr. Newman opened the paper by a grateful and 

 honourable recognition of Mr. Spottiswoode's labours. 



VOL. VIII. 2 K 



1 D E 

 D 1 F 

 E F 1 



and H = 



