298 
MR. R. S. HEATH OH THE DYNAMICS OF 
that is 
cq —cici y -J- -f- cc ^ J rffi J t~99i J rhh 1 . 
This and similar equations give us the formuke for transforming the coordinates of 
a line. The transformation is orthogonal, and proceeding as before, we may use another 
transformation scheme, viz. 
a 
b 
c 
/ 
9 
h 
«o 
cq 
Cl 
A 
9i 
K 
cq 
h 
C 2 
A 
92 
K 
Co 
«3 
h 
C 3 
A 
9s 
h 3 
/o 
cq 
C 4 
A 
94 
K 
9o 
cq 
h 
c 5 
A 
9o 
K 
K 
cq 
c 6 
/e 
96 
h 5 
This determinant possesses properties similar to those proved for the other deter¬ 
minant ; the sum of the squares of the terms in any row, or any column, is unity, and 
the sum of the products of corresponding terms in any two rows, or in any two columns, 
is zero. 
26. From this scheme it follows that the coordinates of the edges of the fixed 
tetrahedron referred to the other will be (cq, cq, cq, cq, cq, cq), &c. There are other 
relations between the constituents of the determinant, due to the fact that opposite 
edges are conjugate polars with reference to the absolute quadric. Thus 
«i=/J h=94 
f\— c h) 9i = h 
If we square the determinant we get 
D 2 =l. 
As before we take D=1 ; in fact, D is the determinant formed out of the second 
minors of A. Whence 
D —A 3 .* 
Hence if we choose A = 1 we must have D=1 also. 
Each constituent of the determinant D is equal to its minor. Also each minor of D 
is equal to its complementary minor. 
27. In any displacement of a rigid body there are always two lines which remain in 
the same position after displacement. 
* Cf. Scott’s “ Determinants,” chap, v., § 9. 
