50 PEOFESSOE DONKIN ON THE EQUATION OE LAPLACE’S FUNCTIONS, ETC. 
result: put, for convenience, sin sin ^=0, sin ^ sin ^ = 0' ; then the coefficient of 
r” in the development of (l — 2r(cos 6 cos & -{■ sin ^ sin ^ cos may be expressed in 
either of the two forms. 
s T n ' flsina')” ^"^' + tan-tan 2 Cos(p+«,(^tan 2 tangj cos 2^ 
/ fl S'\" 1 
tan - tan -j cos n(p j 
-^-7^^-^{Co0”0'”l+Ci0”~'0'”“Ysin 0 sin 0'Y cos ® 
sin S sin 6 )” ^ ^ j r 
1 
(23.) 
(24.) 
+ — +Cj0” "0'” *(sin ^ sin &Y cos ip -\- . . . 4-c„(sin 0 sin 0')-” cos ?z®} j 
and it only remains to determine the constants. 
11. For this purpose we may adopt a process of which the first part is taken from 
Laplace. Let cos 0=z(jb, cos 0'—(jtJ, and suppose the coefficient of 7'"cosi(p in the deve- 
lopment of (l — 2r(|«<|(A'+\/T — \/l — (M^.cos to be itself developed in a series 
of powers and products of (Jj and im'. It is only necessary to ascertain the term inde- 
pendent of (M, jW/', and the term containing the product ; and for this end we may 
write the above expression thus, (1 — cos^)+?^)“l, since the terms g,'- cannot 
affect the required result ; and this again may (for the present pm’pose) be considered 
equal to (1 — 2rcos(p+r^)“i-i-i'^|W/V(l — 2r cos (p+r^)~t; and if we put 2 cos (p=A-f -~5 so 
that 1 — 2r cos <p-l-p®=(l — ?’^)^l-f--^, and develope m the usual manner, we easily find 
the following results : — 
1st, if n—i be even, the term independent of (jtj, (jtj' in the coefficient of z‘"cosz<p is 
Q 1^.32. 5L..(n + f- 1)2. lL3L5L..(n-2- 1)2 
1 . 2 . 3. ..{n + i). 1 . 2 . 3. ..{n—i) ’ 
and the term containing vanishes. 
2nd, if 71 — ^ be odd, the term independent of jW,, jM,' vanishes, and the term con- 
taining (Jtjfjij' is 
l'^.3^...{n-\-i)^.l'^.3^....{n—i)^ i 
1 . 2 . 3 . . . (n + i) . 1 . 2 . 3 . . . (n — z) ’ 
the factor 2 to be omitted in each case when ^I=0. 
12. On the other hand, the coefficient of r” cos zip is (art. 10) 
Ci(sin^sin^')“*0”“*0'”“'(sin^)2'(sin^’)2'; (25.) 
and we may find the term independent of (jb, ptj', and the term invoLlng g-fi' in the deve- 
lopment of this, as follows : — 
It is easily found, from the theorem of art. 7, that 
(sin ^)“'0"~*(sin^)2' = l. 2... (^^—^)(sin^)”X (coefficient of r""‘ in the development of 
(1 — 2rcos^+p^)“^''^®^) ; and for om’ present purpose we may put sm^=l, and 
(l-2rcos^+r2)-('+U=(lq_^2)-(‘H)_|_(2i+l)^r(l-l-r)-(''^^). 
