472 
PROFESSOR A. SCHUSTER OH THE DIURHAL 
Both expressions are included in the general one 
A, = — (2?‘ + 1) m 
{m — {i — 2)] {m — {i — 4)} . . . {m + {i — 2)} 
{m — {i + l)}{m — (i — 1)} . . . {m + (^ + 1)} 
where successive factors of both numerator and denominator increase by 2/' 
If in formula (1) we substitute for any term P, 
djjb djjb 
(2^ + 1) P/ 
(2), 
we obtain an expression for cos mB in terms of the first differential coefficients of the 
zonal harmonics ; and, if in the formula so obtained we successively apply the trans¬ 
formation 
^P+i _ dn\_^ 
dy.'P 
(2^ + 1) 
we finally obtain the required expression in terms of any required differential 
coefficient. 
I have obtained in this way the equation 
cos mu 
(— l)^'^ ' ^ g g -^m+p-Z^^'^m+p-Z + • • • A/CT^P^ + . . . (3), 
wliere cPVi is written shortly for and 
A,- = {2^■ + 1) m 
{m — {i — p — 2)}{m — (i — p — 4)} . . . {m + {i — p — 2)} 
{m — (i + p + - {i + p — 1)} .. . (m + (i + ^ + 1)} 
(•i). 
except for the last term of the series, which will be given presently (5). If p = 0, 
this expression agrees with the one previously given, and, as I shall proceed to show, 
if it is true for any value p, it will also be true for the value p + 1. Assume, tlien, 
the equation (3) to liold. 
The relation 
+ + = {2i -f 1) d^^P, 
shows that the factor multiplying d^'^’P^ + i in the expression for cos mO will depend 
only on Ai and on A^; in (3). If + is this factor, we have 
P _ Aj A; _^2 
V + ’)—2^• + l 2i + rd 
* In tliG very useful book on ‘ Spherical Harmonics,’ by Ferrers, the factor m does not occur in the 
general expression for A; (page 83) ; but, from the deduction of the formula, it is clear that the factor vi 
must be taken twice when it occurs in the numerator, and not at all w’hen it occurs in the denominator. 
In using the equation I was at first led into error by the ambiguity, and hence I believe the expression 
given above to be clearer. 
