192 
MR. G. H. DARWIN ON THE STRESSES 
Then we may put 
W*=o*— 
2! 
4! 
6 ! 
-/rt 6 + . 
• (12) 
w, is a solid zonal harmonic of degree i; but r 1 Wi requires multiplication by a 
factor ( — !} 2 in order to make it a Legendre’s function. 
The factors by which Wi must be deemed to be multiplied in order that it may be 
a potential, will be dropped for the present, to be inserted later. Or we may, if we 
like, suppose that the units of length or of time are so chosen as to make the factor 
equal to unity. 
Now let 
o _i o p p _&(i-2)\i-W c__ 
Po—L P 2 — 2 !» Pr— 4 | > P6 y, 
(13) 
Then, dropping the suffix to W for brevity, we may write 
W=fi 0 p i —/3 2 p i 2 z 2j rfi 4 p l 6 z 6 +.(14) 
1 shall now find P, Q, It, T at any point in the meridional plane which is determined 
by y— 0. 
In evaluating the first differential coefficients of W we must not put y— 0, in as far 
as these coefficients are a first step towards the determination of the second differential 
coefficients. But in as far as these first coefficients are directly involved in the 
expressions for P, Q, It, and T, and in the second coefficients in the same expressions, 
we may put y= 0 , and thus write x in place of p. 
Then 
dp 
x ’ p^r y ’ 
since p 2 =x~-{- y 2 . 
dW 
dy 
dW 
dz 
=x[i(3 0 p; i - 2 — (i—2)l3,p i ^ 2 +(i—4:)l3 4 .p i -h i — . . .] 
=y[same series 
= 2 [ - 2/3^-- + 4£ 4/ T'- V - V "f • • •] 
In differentiating a second time we may treat p as identical with x, because y is to 
be put equal to zero. Thus 
d?W 
d,x % 
d 2 W 
dy 
d?W 
dz 2 
=i{i— 1 )fi 0 x‘~ 2 —(i —2 ){i— 3)/3 2 P“V 2 + (i — 4) (1—5) (B 4 x‘~ (] z: 
= ipyrr 2 — (i —2)/3 2 * <_4 z 2 -f- (i —4)/3 4 £c* _< k 4 — . . . 
— L.2/3 2 £c^ 2 +3.4 / 8 4 ,^ _ % 2 — o.6/3 6 a;' i ~V'+ . . . 
1“ (15) 
