160 The Rev. S. Haveuron on the Equilibrium and 
1 = be” + bc, m = c a’ + ca if n = ab’ + a’b’, 
U =b"c +b", m=c"ateca"’, nn’ =a"b+ ab", 
“= be +0'ce, m’=ca’ +ca, n’= abl’ + ab, 
and, effecting the transformations, we shall obtain the following values for the six 
quantities, = a = U, V, W, expressed as linear functions of the six transformed 
cae d# dh oa ; 
quantities, Ta” dy dy” dz” Py VES Wa 
dé ust » Arf 2 = 
in? at — +e yt b¢.u' + ae.v +abw, 
fie. ce oi +c ote 7 $i cu +a'c.v +a'b'w’, 
dy — dy 
Ts 113.6 gl ae bc". U ‘al cl" v-+- a/b! w’. 
(13) 
v=p= a ute ok + lo! + mv’ + nw, 
dif J U2 Uh) 
ver g+¢@ = 47k 4 tut mv + al, 
/ 
p” p+ eH ees Wo! mv! + nl’ 
The substitution of these values in the function v, will transform it into another 
form, which still retains the twenty-one terms, though now expressed in terms of 
the six new variables; the coefficients of which will be known functions of the 
nine angles which determine the position of the new axes of coordinates with 
respect to the old axes. Our object is to discover relations among these co- 
efficients which will enable us, by satisfying them with real values of the nine 
cosines (a, b, c, a’, b', ce’, a’, b’, ce), to reduce the function v, to a more simple 
form, and consequently to diminish the number of constants, which, in general, 
is fifteen. 
Let (N52 Jv No Do I» Ny Dy 3) denote the transformed values of (4,,8,.%, 
4.5 Bry Yo» 4 BysY;) 3 the values of these coeffieients, when developed, are, 
