164 The Rev. S. Haueuton on the Equilibrium and 
such that a knowledge of any properties of the one must throw a corresponding 
light on the other. This surface is, 
(a0! + ny! + ce!) + 6(1y%2* + mas? + Nay’) (17) 
$4ye(3a,2" By yc") +402(a,0°+38,y" 7.2") +4:0y (250° By +37,2°) = 1 
The relation existing between this surface and the function v, is the following: 
that, if this surface be referred to a new system of rectangular coordinates, the 
new coefficients will be expressed by the same functions of the nine angles as the 
coefficients of the function v,, when similarly changed by a transformation of co- 
ordinates ; so that the relations (15) may be established among the coefficients 
of the surface (17), and vice versa, if we knew any geometrical properties which 
would reduce the surface, there would exist corresponding properties of the 
function v, which would throw light on the nature of the medium represented 
by v,. The relation which exists between the function v, and the surface (17) 
may be proved as follows; the surface may be put into the following form: 
[a(a*) + w(y*)? + e(2*)"] + [-Qyz)’ + M(2a2)° + n(2ay)"] 
+ 2[ u(y") (2) + m(a°) (2) + (2°) (y*)] + 2[a(22y) (222) + B,(2xy) (2yz 
+ ¥5(202) (2y2)] 
(2yz)(a(2*) + By") + n(2)) 
(2az) (a,(2°) + B(y") + 72(2")) 
(2ry) (a,(2°) + B,(y*) + ¥3(2)) 
which is the same function of the six quantities, 2°, y°, 2°, 2yz, 2xz, 2ay, that v, 
: dé dy d 
is of — aa’ 7 “s 
PP =ax?t Py? +02” + be.(2/2') + EAC 2’) + ab(2z'y’), 
yi = alta 4 By? 4 Pe 4 Vel (2y'z) + ale (2x'2’) + a'¥ (22'y), 
ae alr 4 Wy? + 2 40! (2y'z) +c! (2a'2') +06" (22'y'), (18) 
Qyz = px” + qy” + 72” + U(2y'2') + m(22'2') + n(22'y’), 
Qez = pa? tefy? tr'2? 41 (2y'2’) + m'(22'2') + n'(22'y’), 
Day = pla + gy” + r'2® + (2y'a!) + ml(2a’z!) 4 nl! (2a'y'). 
These equations of transformation, having the same coefficients as the equations 
(13), prove, that the transformation of the surface (17) will produce the same 
coefficients as the transformation of the function v,. 
» U,V, Ww. Also, it may be easily seen that 
