ACTING INVERSELY AS THE SQUARE OF THE DISTANCE. 193 
The coefficients A°, A’, A", &c. are functions of u and A, which 
satisfy known partial differential equations. (Resultate, 1838, 
p- 22.)* The same is the case with B®, B’, B’, &c. On the given 
surface the potential must be equal to a given function of uw and 
A, namely, V = U, thus 
1 
(4)? V=(14+y2)?U. 
Now if we assume (1 +2)? U to be developed in a series 
PO ee Pre 2 oe, 
in such manner that the several members P®°, P’, P"’, P!’", &c. may 
likewise satisfy the differential equations in question ; and if we 
bear in mind that both the above series for the potential remain 
valid up to the surface, it is evident that 
; Po + P+ P! + Pl + &e. 
=A°(1+ yz) P+ Al(Lt yz)? +A" (lL + yz)? + &e. 
= B(1 + yz)—?#+ Bi(1 + yz)? + BY (1+ 72)? + &. 
We conclude hence, that if quantities of the order y be neglect- 
ed, P? + P’+ P+ &c. = A°+ A’ + A" + &c.; and thus (as a 
function of u, A can only be developed in one series, whose 
members satisfy the above-mentioned differential equations) 
P°= A°, P’= A’, P’!= A", &c. In like manner quantities of 
the order y being neglected, P° = B®, P! = B’, P” = B", &c. 
Thus if we write (I.), 
; A° = P°+ vy a, =P-—y2 
r A'=P + yd, es P—yi 
* A" =P" + ya", B’'= P! — yo" 
% Al =P" 4 yall, Bll= Pl _ 5 vy", &e., 
‘where it is obvious that @°, a’, a",a!", &c., and likewise 6°, 6’, 6!,b", 
&e., will satisfy the differential equations in question ; and if 
we substitute these values in the above equations, neglecting in 
so doing quantities of the order 7’, then after dividing by y, we 
shall have within errors of the order y, 
Pt+adt+al+a”+ &.=12(P°4+3P 45 P" +7 P+ &e.) 
P+ b+ O44 &e. = 12(P°43 P45 Pl 4 7 PP” +&c.) 
Therefore within errors of the order y, 
P =a, =a, b' =a", &c. 
and consequently, within errors of the order y*, (II.) 
B® = Po — yo, B' = P — ya’, B" = P" — ya", &e. 
* Scientific Memoirs, vol. ii. p. 203, 
