•Vi 
PEOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 
expression becomes 
and if we develope each term and collect the coefficient of we tind that it is 
^ n{7i—\) n[n—\){n— 2) [n—^) n{n—^)...{n—2i-\-\) 
fi 22 "1" 22.4® 22.4^.62... (2z) 2 +•••’ 
a series of which the sum * is ^ ^ ^ ) 
15. It follows that 
1 .2.3...« 
The truth of (27.) is therefore estabhshed. 
0 ”^tan 2 j =0" 'G'^tan^j =1.3...(2^— l)0"“‘(sin^)^', 
and consequently the coefficient of 7•”cos^^ may also be expressed in the form 
1.2.3... + ^ j)‘(tan 1)', . 
( 28 .) 
tile factor 2 being omitted as before when ^=0. 
The law of the numerical coefficients in this case is remarkable. For if the expres- 
.sion (1-f-cos be developed in the form Ao+^iCos (p+AaCOs 2<p-|- • • +-^cosw(p, it will 
be found that 
2 
A.= l + (o^^itting the 2 when i=0), 
so that we may express the law of the development of 
(l — 2r(cos ^ cos ^' + sin 6 sin ^'cos 
as follows : — the coefficient of r” is 
1 
1.2..3...n.l.3.5...(2ra — 1) 
(sin ^)~"(sin d') ” ^sin 0 sin 0^ ^sin 0' sin 0'"^ U„, 
where 
0 fl ' / 0 \ 2 / 0'\2 / 0 \”/ 0 '\” 
U„=Ao+A,cos(ptan2tan2+A2Cos2<p( tan- j Itangl + ... +A„cosw<p( tan^ 1 (tan-j , 
and A^ is the coefficient of cosi(p in the development of (l+cos<p)". 
Although this form of the development follows a more simple and remarkable law 
than that of art. 13, it is evident that for actual calculation it ■would be much less con- 
venient, since the number of operations is much greater, and the differentiations more 
complex. But as the complete explicit form of the development is kno^^m mdepen- 
dently, this consideration is not of much practical importance. 
* — \ 2^ — n — ^ coefficient of x’' in the development of (1— 2a:) ^ ; now 
— = +1(1 — a;)" V + ^ (1— a:)“^a:''+ ... ; 
and if each term be developed and the coefficient of x" collected, the result is the series in the text. 
The equation (27.) probably admits of some simpler demonstration, which I have failed to perceive. 
