452 REroRT — 1882. 



by the rule of distributive mtdtiplication, then this product must represent one and 

 the same result, lohatecer the directions of the components of the factors may he. 



As to details of the way of proceeding, we refer the reader to our paper with the 

 above title, printed in the 'Transactions of the Royal Society of Edinburgh,' vol. 

 xxvii. ; but in order to give an outline of its contents we may state that we con- 

 sider in that paper two sets of expressions for two vectors, p and -or. Namely 



(1°) [P^'^P^P', 



^ \w- Tw\_ Up cos ptsr + U(T sin pw], 



(T being coplanar with p w, and at right angles to p, on the same side as w ; and 



(•~>°\ f P^ *"' +./^ + ^^' 



i,j, k being a system of tri-rectangular unit-vectors of any orientation whatsoever. 

 Forming with each set the expressions 



P = p-ar — ■arpf Q = pw + Tsrp, 



and identifying Pj witli P.,, and Qj with Q.,, we are led to the conclusions : 



(1°) That the expressions Up Ua — Ua- Up, and similar ones in {, j, k, are vectors, 



perpendicular to Up, Ua, for the first expression, and perpendicular to /, k, in the 



case jk — kj, and so on. 



(11°) That the expressions Up Uv H- Uv Up, and similar ones in i, j, k, are 



scalars, and scalars of one and the same value, namely, the value being equal to 



zero. Further deductions present themselves easily. 



The establishment of (C/p)'^=— ], t"-=— 1, Sec, on the contrary, must be 



made dependent on the discussion cf the product of vector factors at least three in 



number. 



6. On Linear Syzygetic Relations between the Coefficients of Ternary 

 Quadrics. By Professor R. "W. Genese, M.A. 



Let {a, h, c,f, y, h){:i; y, s)'- = be a ternary quadric. 



This represents various curves and surfaces according to the system of co- 

 ordinates and the locus in quo. 



To fix the ideas, consider .c, y, z, as the triliuear co-ordinates of a point, so that 

 the quadric represents a conic. 



Now let 



In + nib + nc + pf+ qy + rh = 



be a relation between the coefiicients. 



Then it is known that the conic is such that triangles can be inscribed in it 

 which are self-conjugate with regard to a fixed conic ; but also that if the multipliers 

 be themselves connected by the relation 



= 



then the conic is such that two fixed points can be found which are conjugate with 

 respect to it. Tn this case the conic {I, m, n,p, q, »-)(.r, y, s)'^=0 represents two 

 straight lines, whose poles, with regard to the imaginary conic, .r- + y' + s' = 0, are 

 the two conjugate points. 



If, however, two linear syzygetic relations be given, then these determine two 

 pairs of conjugate points (and a connected pair). For, from the two relations, we 

 can form the relation (X,/ + \'l')a + &c. = 0, and choose X : X', so that the symmetrical 

 determinant is zero in three ways: these are not independent, for the polars of the 

 conjugate points obtained are determined by 



X(;, m . . .)(.r, y, zy + X'(// m' . . .%v, y, zf = 0, 



i.e., they are the joins in pairs of the points determined by (/, m . . .)(.r, y, s)^ = 0, 

 and {I'n ' . . .)(.r, y, z)- = 0, and two pairs of joins clearly determine the third 

 pair. This is Professor Hesse's theorem, viz., conies which divide harmonically 

 two diagonals of a quadrilateral divide in the same way the third. 



