DIURNAL VARIATION OF TERRESTRIAL MAGNETISM. 
199 
does not differ by more than 1 and the degree n by not more than 2 from the type 
and degree of the original velocity potential. 
For the calculation of these special cases the previous investigation furnishes the 
necessary formulae. Confining ourselves to the velocity potentials i/C and we find 
the requisite numbers collected in the following table :— 
Values of e n a +//. 
Velocity potential = 
h 1 - 
Velocity potential = y 2 2 . 
cr = 
i 
2 
cr = 
1 
2 
3 
n = 1 
9 
1 
10 , 
5 7 
n = 2 
7 7 
21 7 
2 
1 
3 
4 
6 
15 
9 
45' 7 ' 
1 
3 
9 
Ay 
140 r 
6 
45 7 
4 
l 
-<ri 
o 
35 7 
Equation (42) holds also for negative values of o-, but when a is smaller than —2, 
it is more convenient to calculate the q coefficients from the values of p/ already 
given. We find for this case 
?»' = 
n. n + 1 
[Jy {/*»' ! + (n+tr+1) (n-o-)/x/ +1 } -cryTC]; 
which, as may be expected, is identical with the equation connecting p and k. 
For (7=1, (42) gives 
p n 1 - + ] [iy A 2 + n(n+l) K n °}+y'fc n x ] 
= \y {(^—1) {n + 2) p n 2 + K n °}—y'p n , 
and hence 
q'n = t [h {(^- 1 ) (n + 2) p n 2 +K n 0 }—y'p' n ~\ .(43). 
n . n + 1 
Our equations are not valid for the case cr = 0 , because they depend on a division 
by a-. The first of equations (14) from which we started containing R only in the 
form dJi/d\. is obviously unsuitable to determine those parts of R which are 
independent of X; we must, therefore, have recourse to the second equation 
dU 
w = p 008 
dd p sin 6 dX' 
