478 
PROFESSOR A. SCHUSTER OR THE DIURRAL 
&c., we have'an equation for that part of Y which depends on cos X and sin X. 
After multiplication with sin u, one side of the equation contains the colatitude in 
form of tesseral harmonics only, and hence we obtain at once the required 
expansion of The equations are 
= cos X 
+ sin X 
24-09 ^ + 35-59 — 8-45 'y*"' + 2-39 ^ — 2-41 — 1-64 
dfju dfjb diJb djjb dfj. dfji 
48-00 + 81-52 + 5-66 — 7-62 — 1-09 ^ 
uiJb clijb clfM dfjL ayu 
dl\ PP,1 
- 5-78 ^ - 1-57 - 0-34 — j- 
ay cty cty J 
With the help of 
we have, finally, 
dY 
dx. 
= Ya sin u, 
- = cos X[48-00TV + 81-52TV + 5 - 66 T 3 I - 7-621V 
— 1-091V - 5-78T61 — 1-57T/] 
+ sinX[ - 24-09l\i + 35-59TV + 8 - 45 T 3 I — 2-391V 
+ 2-41TV + l-OlTgi + 0-34T/] . . [A], 
To obtain, similarly, an expression for the series of coefficients h.^ were 
expi-essed in terms of sin u, sin 2u, &c., giving the equations 
O 2 = 67-0 sin u + 6 I -3 sin 2a + 7-4 sin 3a — 11-6 sin 4zt + 15-9 sin 5a + 18-6 sin 6u, 
= 21-5 sin a + 113-2 sin 2u -j- 7^7 sin 3a — 12-6 sin 4 m — 10-8 sin Sa — 28-2 sin 6a. 
From the known expansion of con pu, in terms of the first differential coefficient of 
the zonal harmonics, we can obtain by differentiation an equation which will at once 
give us in the required form. 
Thus, for instance, we have by Equation D, page 474, 
cos 4 a == 
(/P, 
dfjb 
] 6 
4 5 
dV r/P 
u i. 3 I 3_ 'ihi 
and differentiating with respect to p. 
4 sin 4 m / 
sin a = 
dp3 
1 6 
4 5 
d^>, 
dll^ 
, 3 _ 
3 5 
dfl'' 
By substituting this and other similar expressions, we find 
