162 The Rev. S. Haucuton on the Equilibrium and 
N, = (aa°% + Bab! + ca’*b”) 4+ 3(Lna‘a” + un'aa” + nn''aa’ ) 
+ 3a,(na>+pab) + ,(na* + pa'b’) + %,(na'? + pa’b”) 
+ a,(n'a? + p'ab) + 3B,(n'a? + p’a'b’) + (n'a + p'a’'b"’) 
+ a,(n"a” + p’ab) + B3(20a” + p’'a'b’) 4- 3y,(n"a'” + p’a"'b") 
3, = (ab ’a + Bb? a’ + cb’’a"’) + 3(Lnb/b" + mn/bb” + nn''bb’) 
+ 3a,(nb’ + gab) + B,(nb? + qa'b’) + y(nb’? + qa"b’) 
+ a,(n'b? + q'ab) + 3B,(n'b? a q'a'b’) = y(n'b’? + gab’) 
+ a,(nl'B? + q'ab) + B,(nl"b” + q/a'b’) + 3y,(n!/b" + q'a'b") 
), = (ac’ab + Be?a'b! + ce!?a’b") + 3(une'e” + mn'ce” + nn'cc’) 
+ (a,(7'n"! + rn’) + B(7r"n+rn”) +y,(rn' +r'n))— (Lab + ma’b’ + nab”) 
+ a(ne>+rab) + 8,(ne?+ra’b')  +4,(ne'? + ra’'b") 
+ a,(n'c + r’ab) + B,(n'c” + r'a'b’) + Yo( nc!” + TOO a) 
+ a,(n''e? + rab) + B,( ne” + rab’) + ¥5( n''¢!” +. r'al'b"’) 
In obtaining the foregoing values of the coefficients, use has been made of the 
following relations among the nine angles: 
a='e” —b'e', a =b"c—be"’”, a’ =be’— be, 
b= cla! N00) 9 bs h0da 00s ee eC eo. 
C010 = 00 ee Ch ee Ce 
which may be easily seen to be identically true. 
It can be proved from the values of the nine coefficients which have been 
P 
given, that there exists a real system of rectangular axes at each point of the 
body, for which the following equations are true, 
Nie ev ny Pe = 0) tas Has; (15) 
for, we obtain from (14), since, p+ qg+r=0, p'’+q'+r'=0, ke. &e. 
Ni t2:+3):= (4—x)be +(B—m)b'c’ + (c — N)b"c” 
+ (a +Bi+ 1) + (a2 + Ba + V2)!" + (4, + B+ 2) 0": 
N. + 2.+), = (a ~ L)ac + (B —m)a'c’ + (c — nae” 
+ (Gq +B, +11)m + (+B. +%2)m! + (45+Bs +75)" 
N; +2; +9,= (a— )ab+ (B — M)a’b! + (c — n)a’b” 
+ (G4 +h +1) + (4,+, +2)’ + (4, +B,+7,)n”. 
