PEOFESSOE DONKIN ON THE EQUATION OF LAPLACE’S FUNCTIONS, ETC. 55 
or, reducing as before, and observing that is equivalent to 
we have, finally, 
(s“^)'“(sir9 (s) sISl) •®' 
• m 
If in this expression we put 0 for the result of all hut one of the operations indicated in 
the last term, we reproduce the form (21.) obtained in art. 9, which appears in fact to 
be always complete. The form (34.), however, as before remarked, wiU never contain 
superfiuous constants. 
20. It is an interesting question whether the forms thus obtained on the supposition 
that is an integer, are not reaUy general, inasmuch as they are expressible without 
reference to that supposition. I beheve that they are ; but I doubt whether, in the 
existing state of analysis, it can be proved either that they are or are not (see ^ IV.). 
21. It is easy to obtain other forms analogous to those given in the preceding articles, 
containing inverse in place of direct operations. But such forms are objectionable, 
because they necessarily give rise to superfluous constants, the discrimination of which 
may be a matter of difficulty. 
Section IV. 
22. It was observed at the beginning of this paper that the equation 
d'^u d'^u d'^u „ 
dx^ dy’^ dz^ ’ 
when transformed to polar coordinates, may be written in the form 
{(sin + (D'+Csin «)v|(4+l)}w=0 (36.) 
I do not know whether this form has been noticed ; but it is remarkable in this respect, 
d 
that if the symbol be replaced by n, it becomes identical with the equation discussed 
in the fii’st section, and of which the solution was shoAvn to be 
(sin ^)“”^sin ^ ^ sin “1"^ tan (36.) 
It is not easy to verify, by direct substitution, that this expression satisfies the differential 
equation of which it is the solution ; at least I have not yet succeeded in doing so ; and 
even if this were accomplished, it is most likely that difficulties would remain to be 
overcome before we could estabhsh the legitimacy of extending this result to the case in 
which w is a symbol of unlimited signification. Still it seems worth while to try, at 
least as an experiment, the consequence of assuming that the form (36.) will give the 
solution of (35.) on putting instead of n. 
I 2 
