OP UNRESTRICTED DEGREE, ORDER, AND ARGUMENT. 
523 
referring to the formulae (40), (50), we see that, as special cases of this result, 
P/ +2 (fi) + 2 (m + I) ^ , P n m+1 (n) — (n — m) (n + in + 1) P n m (/*) = 0 ! 
^ H126), 
Q/ i+2 (/*) + 2 (m 4- 1) yy* Q** +1 (p) - (n ~ (n + m + 1) Q/ z (/a) = 0 j 
the formulae (125), (126), are well known for the case in which on and n are real 
integers. 
If /x = cos 6, then introducing the modification of Art. 17 into the symbols P,“, Q/', 
we have 
P/‘ +2 (cos$) — 2(m+ l)cot$. P/ 2+1 (cosi9) + (n — m)(?r + on -f l)P,/"(cos0) = O 
Q„ m+z (cos6) — 2(m+ l)cot0.Q a *" +1 (cos0) + (n — m)(w + on + l)Q/'(cos0) = O ^ ^ 
Toroidal Functions. 
56. If A, B are points at the extremities of a diameter of a fixed circle, and the 
coordinates of any point P in a plane through AB perpendicular to the plane of the 
AP 
circle, are denoted by cr, 6, where <x = l°ggp > $ = A APB, and <f> is the angle 
the plane APB makes with a fixed plane through the axis 0 z which bisects AB and 
is perpendicular to the plane of the circle, it is known # 
that the normal functions requisite for the solution of potential problems connected 
with the anchor ring are 
T-. , , . COS n cos 
Pu-F (cosh a) . nO . rrub, 
■ v ’ sm sm r ’ 
Q / , . cos ^ cos . 
(cosh a) . nd . ond>. 
- v ' sm sm 
* See C. Neumann’s ‘ Theorie der Elektricitats- uud Warme-Vertheilung in einem Ringe,’ Halle, 1864. 
W. M. Hicks, “Toroidal Functions,” ‘Phil. Trans.,’ 1879. A. B. Basset, “On Toroidal Functions,” 
‘American Journal of Mathematics,’vol. 15. W. D. Niven, “On the Ring Functions,” ‘ Proc. Lond. 
Math. Soc.,’ vol. 24. 
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