46 PEOFESSOE DOTOQN ON THE EQHATION OF LAPLACE’S FTJNCTIONS, ETC. 
This relation is true whatever be the value of n. But as we are now supposing n a 
positive integer, it is evident that we have 
— ( 12 -) 
where is given by (6.) or (7.) of the last article in the general case, and by (5.) in the 
case in which is a function of d without (p. 
This solution contains two arbitrary functions in the former case, and two arbitrary 
constants in the latter. It is therefore, in general^ the complete solution. (See, how- 
ever, arts-. 8, 9, 17, &c.) 
3. The operative symbol . . .ts-atn-j, wiitten at length, is 
^sin^J-i-wcos^^^sin^^-f(w— l)cos^^ — ^sin cos . . . . (13.) 
Now it is evident that 
sin 4 +»» cos «= I (sin «)’ 
(the subject of operation being of course omitted on both sides) ; and if tMs be put in 
the form 
it will be easily seen that the expression (13.) is equivalent to 
(“■) 
which may also be put in the less symmetrical form 
Either of these forms has the peculiarity, as compared with (13.), of being intelligible 
without supposing n a positive integer (setting aside the difficulties of general differen- 
tiation). 
4. The results of the preceding articles may be summed up as follows : — 
The complete solution of the equation 
l^sin^^g^ -i-7^(»^-^-l)(sin ^)^|?q,=0 is 
7«^=(sin ^)“”^sin sin 4- Cg log tan ; (15-) 
and the complete solution of the equation 
{(sin^^) +(^^) +77(77+ l)(sin^)^jw„=0 is 
i7„= (sin^)“”^sin^^sm^^ |/^e^'^'tan0+F^e~^'^*tan^^|. . . (16.) 
It may be observed that (15.) is deducible from (16.) by taking y’(a’)=Ci+-|C 2 log^r, 
F(a:)=^C2loga:. 
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