54 PROrESSOE DOISTEIN ON THE EQUATION OE LAPLACE’S ECNCTIONS, ETC. 
and so on, till we have, finally, 
— (31.) 
and — ^®)2=0 (32.) 
The point to be observed is, that every value of z which satisfies (32.) will give a value 
of Un satisfying (30.); hence, although the complete value of z will contain w+2 con- 
stants, it is certain that n of them will be destroyed by the n direct operations of the 
expression (31.). In general^ the two constants left are those introduced by the inverse 
operation (do — iii the exceptional cases noticed above, one of these disappears, 
and one of those introduced by the other inverse operations remains instead. 
18. The complete value of in its most general form, is therefore 
— '^^2 ' • • ■ ' • 0 (33.) 
d . 
It was shown before that 7E7„t!r„_i...d-i=(sin^) "|^sin sin ; in hke manner we 
( 1 / d\~^ 1\” 
sml(^) smfl) latter operation being the inverse of 
the former. 
2 . \ -I _ g 
, if ^=logtan^; and this 
may be resolved into either of the two following forms ; namely, 
(l±*) (1+*) • 
The former, omitting the useless factor gives 
the latter, taking the lower signs, gives 
Either of these expressions is to be substituted in (33.), and thus the complete value 
of Un is obtained by means of ^^-|-2 integrations, and n differentiations. In general the 
i^J sml) ^ useless, so that Ave may 
substitute 0 for this expression, as was done in the process first giAen, AAithout ultimate 
loss of completeness. 
19. If we suppose i an integer, and less than ?i, we may stop the process of art. 17 at 
an earlier stage, thus 
'^n 1 • • + 
and 
^n'^n—l • • • — 0 , 
U„ 
:C7„7t7„ 
so that we noAV have 
