PROFESSOR G. H. DARWIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
469 
The equations are tlien 
xb, (^T'' or P^') = 0 , 
X.,-((!r'^ or or or S') = 0 
(5). 
§ 4. 2'h.c Forms of the Functions. 
It is well kiio'\’soi that the function U; is a linear function of /q of degree i made U}) 
in one of the eight following ways :— 
1. When i is even, a linear function of Uf of degree gh 
2, 3, 4. When i is odd, a linear function of of degree T(i — I), multiplied by 
if, or or Cg. 
5, G, 7. When i is even, a linear function of ;q“ of degree T(i ~ 2), multiplied by 
BjCg, or CgAi, or A^B^. 
8. When i is odd, a linear function of of degree 4(b" — 3), nndtijdied by 
A,B,C, 
These eight classes may be conveniently specitied by the initials 0, A, B, C, BC, 
CA, AB, ABC, but it is better to rearrange them according as they are associated 
with the evenness or oddness of i and s, and with the cosine or sine functions. This 
new grouping may be defined by a shorthand notation involving the initials E, () 
and C or S, Avhicli shall denote successively the evenness or oddness of i and s, and 
cosine or sine. 
We shall see below that this arrangement is as folh.nvs - 
(4 or EEC ; 
AB or EES ; 
A or OOC ; 
B or OOS ; 
C or OEC ; 
AB( * or ()ES ; 
rw or ECC; 
CB or EOS ; 
i even, 6" even, cosine. 
I even, s even, sine. 
i odd, s odd, cosine. 
i odd, s odd, sine. 
i odd, s even, cosine. 
i odd, .9 even, sine. 
i even, .s- odd, cosine. 
i even, .s- odd, sine. 
Since the several functi(ms'are linear in uf they are in the new notation lunctiojis 
(jf oi" f, or of a" — 1 and f — 1. 
Hence and involve linear functions of — 1 ol various degrees 
multiplied by various factors ; and the same is true of the functions of [x. 
In the case of the third root the linear function of powers of cos 2(/) may be 
replaced by a series of cosines of even multiples of (j). Further, in forming the C, .S', 
C, S functions we mav regard Ao as being cos (f^, lb as sin (p, and If., as (1 — /3 cos 2(/>)-q 
