VARIATION OF TERRESTRIAL MAGNETISM, 
or, by substituting Aj from (4) and from the corresponding equation, 
{m — — p — 2)} {m — (t — p — 4)} ... {m + (^ — p — 2)} 
473 
Bi+i = m 
[m — (i + p + 1 )} {to — — p — 1 )} • . . + p + 1)} 
1 — 
The square bracket reduces to 
_ ft — pf — (i + p + 3)^ _ 
{m — (^■ + p + 3)} {?«- + (^: + p + 3)} 
so that 
cos md 
_ (_ ly + i - 
'' ’ 1.3.5 . . . (2p + 3) 
where 
= A 
{m — {i — p)} {m + {i - p)] 
{to — (7 4- p + 3) } [m + {i + j) + 3) }_ 
_ {2i + 3) (2p + 3) _ 
[m — (i + p + 3)} {m 4- 4- p + 3; }’ 
1 dP^^^rn+p+l + • ■ • . . . , 
Ai+i = m (2f + 3) 
{to — (-i — 79 — 2)} {to — (^ — 79 — 4)} ... {to 4- ("i — p — 2) I 
{to — ( 7 : 4- 75 4- 3)} {to — (i 4- 79 4- 1)1 ... {to 4 - 4- p 4- 3)}’ 
which agrees with (4) if 73 + 1 is written for p, and t + 1 for i. 
The end terms require special consideration. 
As only the even or only the odd zonal harmonics enter into any one series, the 
difference between m + p and ^ in the equation (3) must be even ; hence, the 
numerators in the fractional expression of (4) consist always of even numbers, what¬ 
ever values m, i, or 7 ) may have. 
If [m + V) be odd, the zonal harmonics will all be odd, and the last term of the 
series will depend on + i if 73 be even, and dPVp if p be odd, for the differential 
coefficients of the zonal harmonics of lower degree will vanish. The expression (4) in 
neither case gives the correct factor, and we must substitute for it in both cases 
(27 + 1) 
{m. — (t 4- 73 4- 1)} [m — (i 4- 7 ? — 1)} . . . {to 4- 4- 73 4- 1) } 
( 5 ). 
If in ( 4 ) i is put equal to [p + 2), the first and last term will be equal to m; in that 
case the m is only taken once. 
The expression (5) is easily proved, and shown to hold also if (m +7?) be even. 
The first term of the series is included in the general expression (4). 
I have deduced with the help of the equation (3) the following relatiojis, which will 
be used in this paper ;— 
1 
cos u — -i- 
^ dp. 
cos 2?^ — ^ _ -9- 
COSZM- 15 15 
(A), 
(B), 
3 1> 
MDCCCLXXXIX.-A. 
