50G 
PROFESSOR G. H. DARWIX OX ELLIPSOIDAL PIARMOXIC AXALYSIS. 
makes the succession of P’s alternately positive and negative when t is even. As 
this is ])ractically inconvenient 1 now define 
(1 - 
2* 
d 
dfjL 
i + t 
^ )\ 
and then retaining the former meaning for the q coefficients, we give the following 
definition— 
= Pif*) + + 
with a similar formula for P'^ (/x). 
Thus we need only remark that in the functions of g the qs corresponding to odd 
])owers of ^ enter with the opposite sign from that wliich holds in the functions of p, 
and the wliole of our preceding results are true with this definition of P- (/x). 
If Pq defines the ellipsoid to wliich the surface harmonic applies, we require the 
expression for the perpendicular on the tangent plane at Pq, /x, (f), and that for an 
element of area of the surface of the ellipsoid at the same point. 
By the usual formula 
/d 
2d 
+ 
“ 1 “ 7 2 
7.2 2 ^ 7,'=^ / 2 1 \2 r 7 2 o 
(V - 1 —«) '• (V - 1 ) * V 
_ (1 -/3)(/>^- M-l) 
(1 + 
» , fL- -1 . „ /-(I - /Scos 2(f.) 
cos- 0 si.r ^ + - 
(M) 
Let J/ii, dm, df be the three elements of the orthogonal arcs corresponding to 
variations of p, /x, (f) respectively. 
Then by the formula at the end of § i. 
dm Y 
Jdd/j,) 
t -'^-Y 
/ o o\/o 1— ^COs2(/)\ - 
(V - dd{’'n- - - _ 1 
70 o/ o 1 \/ l+l3\ 0 
- Dlv - |—j) j'- 
I — /3 cos \ / 1 — 3 cos 2r/> 
1-/3 A 1 - ^ 
-/d) 
1 — ^ cos '-(p 
(50). 
Therefoie 
(50), 
