336 MR. S. S. HOUGH ON THE ROTATION OF- AN ELASTIC SPHEROID, 
or since |"| ' xz dm^xz dm = 0 and v, w, £ are each proportional to e ,w , 
h x = i\ jj | ( ivy — vz ) dm — pw [ J ixz c/S, 
h x — — A 3 j" j (ivy — vz) dm — pcoik | tpcz c/S. 
h 2 = iX [ j'j ( uz — ivx) dm — paj (j t,yz c/S, 
h 2 = — A 2 (wy — vz) dm — panX j £yz c/S. 
Similarly 
(30). 
The equations of angular momentum are 
Jly - hn(0 =: 0, h% T" hy(0 0. 
Replacing h x , h. 2 , h x , h 2 by the values we have just found we obtain the following- 
rigorous equations 
— A 2 [ j" (ivy — vz) dm — pcaiX |"j tycz c/S 
— i\(o |j" (vz — ivx) dm + pad |j £xz c/S = 0, 
— A 2 j [ [ (uz — ivx) dm — pcoik [j tyyz c/S 
+ iXa) j j j" (ivy — vz) dm — pax j1 ^xz c/S = 0, j 
Now by using the approximate forms (28) and denoting by M the mass of the 
spheroid, we have 
| [ (ivy — vz) dm = [jj { — xyO, + (if + z 3 ) 6 X } dm -f fj [ (w x y — v x z) dm 
— |-M a?6 x + terms of order e, 
(uz — wx) dm = [jj*f (A + ad) @2 ~~ xy0 x } dm -f | j | (u x z — w x x) dm 
= fModd, + terms of order e. 
Also from (29), 
\\ & * = t (tttj) f f y° z ~ (rAv,) • 
j ( tpcz c/S = — -rsTTa 5 9. 2 ~± e y e > 
the errors being of the order e 2 . 
