3S0 
PROFESSOR G. H. DARWIN ON FIGURES OF 
negative degrees with respect to the origin 0 as solid zonal and tesseral harmonics of 
positive degrees with respect to the origin o, and vice versd; moreover, the results 
will have to be applied to a sphere of radius a with centre o, and to a sphere of 
radius A with centre O. This last clause is introduced in order to explain the intro¬ 
duction of the symbols a, A, in this place. 
Fig. 1. 
The formulae required will be called “ transference formulae,” because they are to be 
used in shifting the origin from one point to the other. 
The obvious symmetry of our axes is such that every transference formula from 
O to o has its exact counterpart for transference from o to O ; thus a second 
symmetrical formula with capital and small letters interchanged will generally be left- 
unwritten. When necessary, 6, </>, will be written for co-latitude and longitude with 
regard to x, y, z ; and ©, <b, for the same with respect to X, Y, Z. 
Then, since 
R~ = r 3 + c 3 — 2 rc cos 0, 
we have the usual expansion in zonal harmonics 
The usual formula for the derivation of the zonal harmonic of negative degree 
i + 1 from 1 jII is 
(-)* 1 _ ir, 
i\ d/J R i ? 2i+1 ' 
( 3 ) 
Hence, on differentiating (2) i times with respect to Z , or, which is the same thing, 
with respect to — z, we have, from (3), 
But 
Hence 
_E± — 1 * V° i?L V ' J 
R~ i +1 ~ i\ k = 0 dz' c k 
d 
dz' 
kl 
. w k = k (k — 1 ) . . . (k — i + 1 ) Wk-i = jr—:- w*- 
Wi 1 * = « k\ v: k . 
