974 
MR. MACQUORN RANKINE ON AXES OF 
1 8. Of Oblique Coordinates and Contraordinates. 
As there are, in the relations between two systems of oblique coordinates, or 
between a system of oblique coordinates and a system of rectangular coordinates, 
six independent constants of transformation, it is possible, by referring the equation 
of the Biquadratic Surface (19.) to Oblique Coordinates, to make the six terms vanish 
which contain the cubes of the coordinates. 
The conception of the physical meaning of such a transformation is much facili- 
tated by the employment of a system of three auxiliary variables, which will be desig- 
nated as Contraordinates. 
The relations between coordinates and contraordinates are as follows : — 
Through an origin O let any three axes pass, right or oblique. Let R be any point, 
and let Tvd 
UK=r„ 
Through R draw three planes, parallel respectively to the three coordinate planes, 
and intersecting the axes respectively in the points X, Y, Z. Also, on OR, as a dia- 
meter, describe a sphere, intersecting the axes respectively in U, V, W. Then will 
OX=^, 0Y=3/, OZ=z 
be the coordinates of R, as usual, and 
OU=?^, OV=^;, OW=m; 
its contraordinates, being, in fact, the projections of OR on the three axes. 
For rectangular axes, coordinates and contraordinates are identical. 
Coordinates and Contraordinates are connected by the following equation : — 
r^=ux-\-vy -{-wz (34.) 
In the language of Mr. Sylvester, a system of Coordinates and the concomitant 
system of Contraordinates are mutually Contragredient ; and the square of the radius- 
vector is their universal mixed concomitant. 
Let the cosines of the angles made by the axes with each other be denoted as 
follows: coS 3 /Oa=Cj ; cos 20 a? = C 2 i cos.rOy = C 3 ; 
then the contraordinates of a given point are the following functions of the coordi- 
nates : — 
Also let 
U—X d-Cgl/d-CaZ 
v=c^x-\-y -{-c,z 
iv~C2X-\-Ciy-\-z 
(35.) 
1 1 
C3, 1 , 
= l — c‘l—cl—cl-^2c,c^c,=C ; ' 
2 — 
1 —h . 
c “ 
1 — 
c 
c 
Cci ”” 
S'-} 
c 
=h 
1-cl 
c 
Cq CjCg 
— he. 
C 
= ka 
