124 
PROFESSOR C. RIVEN ON THE CONDUCTION 
/ £2 _i_ \ ° o \d?n rm Mdn m w 
(t-+-K~c-)^+U Tf + j I( — ¥ -n= 
(x 2 c 2 +^)n+/jn 
or, more conveniently, 
„cim , n jtn , „ 7 , . . Jd~D. , 1 dD. 
+ 2 f5F +fn - to+x<, (^+FH' 
m- . 
•n-po)= o 
• (19) 
It will be observed that these transformations are suggested by the consideration that, 
when c=0, the equations in 3 and fl should reproduce the corresponding equations in 
£ ; 
r m n and R ; , for a sphere; r- is the semi-axis minor of the confocal ellipse, and we shall 
X 
Yj 
use - to denote in like manner the semi-axis major. 
G. I shall now show how equation (18) may be satisfied by a series of associated 
functions of the order m. If we compare the functions P J l+l , P,/, P,/ -1 , in which the 
constant multipliers are so chosen as to make the coefficient of the highest power of v in 
their rat ional factors unity, it is known that 
•2 _ o 
,,p »_p n+ 1 I — —T> n- 1 
l/± m — L m “l l m ’ 
and hence 
•:p ; p «+-?_i_ - )/r '^- w ' ~ nl ~ ^ p ni~)(n _1- m~) p n _ 3 
^ til 1)1 I / f) -J \ /Q I O \ 1)1 I /A O 1 O 1 \ 1)1 
(2n — l)(2?i + 3) (4;r - l)(4.?i — V — 1) 
• (20) 
It appears from this that the last term will vanish both when n—m and when 
n = 1 + m, the theorem reducing then to 
and to 
,,2 p m —p »+2 i _p in 
w L m — 1 »/ T , i i o >» ’ 
Am + o 
3 
,2p ?« + l— p m + 3 I _p m +1 
m — J- m I - _ J- 
2m + 5' 
It is clear, generally, that w'P,/ can be expanded in a series of associated functions 
of an even or odd degree, according as n is even or odd. Bearing in mind that 
o-->S- 2 4-i f+ii 
P m — — (n[n +1) — k)~P 
we can obviously satisfy (18) by the expression 
