PRODUCED BY ALTERNATING ELECTRIC CURRENTS. 
305 
w = t s— [(M„ cos n0 + N» sin n0) sin pt + (Q» cos nO + R„ sin nd) cos 
1 ZiTTCt 
thus, the currents may be determined to any degree of closeness. 
12. In order to find the couple that acts on the a , b shell, we use our previous 
result (vii.), 
7 rap an* 
D„ ' 
['Q„'N„ - 'M.'RJ. 
As far as terras of the sixth degree in a, a' this is (writing 6 2 + e 2 = p 2 , 6' 2 + c 2 
= />' 2 ) 
+ 
irapcr 
^T 
8irapa 
Do 
2a& 
«a 
? I + ^2 Q i 
aa 
d 2 " 
a. 2 . Zr — c 2 t era’ vr/ . 
-T 5 1 ’ d? 1 
N\ 
+ 
2 ac T aa' -p , 
7 1+ # K 
era' 
, r/ 2 bca 2 ~ 
~dF 
M > • / f 
5 
remembering that the third harmonic terms are (to this order) in the same phase and 
give no couple ('M 3 = 0, 'N 3 = 0). 
This expression is 
7 Taper act 
"1)7 ‘ 
/ 2 db 
\ P 2 
N'r + ^ M',) 1 + ~ (Q'rN', - M'jR'j) 
+ l '- 7 T “ [(5 2 - c 2 ) N', + 2 be M'J. 
L> 2 Cl' p 
Now, — M'jR'j has to be calculated to the second order only, and on reference 
to the values of these coordinates to the first order it will be seen that it is zero. 
Within the first bracket M\ and N\ are required to the third degree. We 
have (viii.) 
2iraper' 21 a'b' I67r 3 a 3 a /2 p 3 & 
w, = - 
N' 1 = + 
D\ P ' 2 + D \T>\<Pp* 
2'ira'pa 21 ct'c I67r 3 a 3 a' 2 p 3 e 
I (a cr etc ), 
-I. It hv w A/ V -B- / f . / \ 
P ' 2 - nw ■ a + a(r )• 
Hence, for the couple we get 
irapcj aa' 2 a T9 
157 l 3 7 1 
+ 
L D'l P 
8irapa a 2 a' a 1 1 2'n-a'pa 21a' 
2-na'pG 2a’ n , 7 . 167r 3 a 3 a ,2 p 3 . , , W7 7 x 
(5 c - 6c) + (« <r + efer ) (6c - 6c) 
DiDvzy 
—— [— 26c . b' + (& 3 — c 3 ) c]. 
D 2 d* P i T>\ p 
This consolidates finally into 
-j-o 87 T 2 p 2 G(j'CO?a ,% r 1 
■*- * 0/0 7 
py'U 
+ 
4a. 3 (p' 3 - d 2 )' 
. . . . (ix). 
L^W ' D , iD 2 P 2 ^ 2 
13. It will be noticed that the more important term is symmetrical, except that it 
MDCCCXCII.—A. 2 R 
