Potentials,  JEolotropic  and  Isotropic.  585 
deal  with  individual  terms.     We  set  out  from  the  sphere, 
for  which 
=  -  1  R  1     I  ^  ^  sin  #  *"  cW  d<t>  (»'  sin  #  cos  *)2v  0*  sin  #  sin<£)2^  (r  cos  602"3 
Tojo    JO  JO 
1.3...2v1*-1.-1.3....2vs-1.1.3...2fs-1 
=  3R2"i+&2+2"3  x —  ; 
1.3...  2^  +  2^  +  2^  +  3 
hence  for  an  ellipsoid  referred  to  principal  axes 
ax2  +  by2  +  cz2  =  1,  with  A  =  bc,  ... 
-  •■>  /A  V>  /  B_  Y2  fCLY3  x  1-3...2i^=T.1.3...2^I .  1.3...2r8-l 
~°VW    VA«/     \AJ  1.3...  2^  +  2^  +  21/3  +  3 
Therefore,  if  2s=2j/j  +  2v2  +  2v~, 
~  (25  +  1)  (25 +  3)^^  ^  |i/3  U«  /    \A«/    VaJ 
3  /AP  +  Bms  +  Ora' 
(2s +  1)  (25  +  3 
/A^+w  +  wy 
)V  A„  /' 
Now  suppose  a'  y  z  transformed  by  a  linear  substitution  to 
other  axes  xx  yx  z1  which  are  not  principal  axes  of  the  ellipsoid, 
and  Imn  transformed  by  the  inverse  substitution  to  l^m^n-^ 
(cf.  Salmon,  Higher  Algebra,  p.  102).  The  transformation 
makes  la?  +  mv  +  nz  =  l1a?1-{-m1y1  +  n[z1,  leaves  Aa  unaltered, 
and  gives  A:Zi2  +  ...  +  2A1W1 ...  for  AZ2  +  B>n2  +  Cn2. 
Hence  dropping  the  subscripts  we  have  the  general  form 
ifc 
+  »'*  +  "-~)»<fr=  (2,  +  lX2,  +  3)ft)-      •      (U) 
In  order  to  deal  with  specified  products  of  inertia,  a  mode  of 
writing  the  coefficients  in  the  expansion  of  a  ternary  quadric 
is  required,  say 
2s  =  vl  +  Vo  +  i>3  y 
Phil.  Mag.  S.  6.  Vol.  11.  No.  64.  April  1906.         2  Q 
with  2s  =  vl  +  Vo  +  i>3  f ' 
