444 
DR. E. W. HOBSON" ON" A TYPE OF SPHERICAL HARMONICS 
required, in which, although n and to are still real integers, /a may have values 
which are real and greater than unity. The functions for which n is fractional or 
complex are required for the solution of potential problems in which the boundary 
consists of coaxal circular cones and of spheres with the centre at the vertex of the 
cones. For potential problems connected with the anchor-ring functions are required 
for which n is half an odd integer, and p is greater than unity. For the space 
bounded by two spherical bowls with a common rim, solutions in which n is complex 
of the form — \ + Ph and p is greater than unity, have been applied. Solutions 
in which to is not an integer are sometimes of use, for example, in the potential 
problem for the portion of an anchor-ring cut off by two planes through the axis of 
the ring, which are inclined to one another at an angle not a sub-multiple of two 
right angles. 
The expressions 
V sin mC t ‘ Un W’ r n sin m $ - U,r 
in which up (p) represents any particular integral of the differential equation which 
it satisfies, and in which the degree n, the order to, and the argument p may have 
any real or complex values, are a special type of Spherical Flarmonics in the extended 
sense of the term, which applies to ail solutions of Laplace’s equation ; the investi¬ 
gation of their forms reduces to that of two particular integrals, here denoted by 
TV” (p-), Q« ra (p), of the differential equation which up (p) satisfies. The forms and 
properties of the functions required for various potential problems have been investi¬ 
gated by various writers, the investigations resting usually on a more or less inde¬ 
pendent basis; thus, for example, we possess separate theories of Toroidal functions, 
Conal functions, &c. It is obviously desirable that all these special functions should 
be treated as parts of a general theory ; thus an investigation of the forms and 
properties of the two functions Pp (p), Qp (p) for unrestricted values of n, m , p is 
required for the consolidation of the various special results which have been obtained 
in connection with special potential problems. To do this by means of the modern 
methods applicable to linear differential equations is the object of the present memoir. 
In the standard treatise of Heine, the forms and properties of the functions P/‘(p), 
Q / 2 (p) are investigated for complex values of p, the degree n and the order to being 
primarily real and integral ; various extensions are made to cases in which n is not 
so restricted, but in default of a general definition of the functions for unrestricted 
values of n and to, these extensions are fragmentary, incomplete, and in some cases 
erroneous. Many of the series which satisfy the differential equation for unrestricted 
values of the degree and order have been given by Thomson and Tait,* and a general 
treatment of the series has been given by Olbricht,! who obtains seventy-tw r o hyper- 
* See ‘ Natural Philosophy,’ vol. 1, Part I., Appendix B. 
f See Olbricht, ‘ Studien iiber die Kugel- und Cylinder-functionen,’ Halle, 1887. 
