PEOEESSOE CAYLEY ON PEEPOTENTIALS. 
755 
Hence 
r(js + g) 2CvJ/ 0 ( e 2 
5 o„is' 
r | 2 c^o_A 2 y 
27riT(g + 1) ’ A 0 /<?. . .h \ \) 
__ - r (^+ g ) 2C4 / 0 a» f 
2^V(q + 1) 'A 0 fy. ..h \ p / " * AV ’ 
where \J/ 0 , A 0 are wliat \|/, A become on writing therein k=0. It will be remembered 
that A is what H becomes on changing therein d into X ; hence A 0 is what H becomes 
Moreover \|/ is what <p becomes on changing therein a, (3 ... 7 into 77 . . . £ : writing 
X=0, we have il= g . .. £=|; hence \I / 0 is what <J5 becomes on changing therein 
a, (3 . . .y into . P. And it is proper in <p to restore the^ original variables by 
writing , ■ X — ^ in place of a, fi . . . y. 
v7 2 +fl V/+0 V/* 2 -M / 
127. Recapitulating, 
V=' 
qdx . . .dz 
\{a-xY... + {c-zY + e^ 
where, since for the value of V about to be mentioned g vanishes for points outside the 
ellipsoid, the integral is to be taken over the ellipsoid 
— +— =1 
and then (transferring a constant factor) if 
T- M+D uf h) H .r , 
v r(is+ ? ) -A.U “fjj T » Vi+/ s ...(+s* 
the corresponding value of § is 
where A 0 is what H becomes on writing therein ^=0, and \f / 0 is what 4 1 becomes on writing 
^ ~ in place of a . . y. 
f ' l 
128. Thus putting for shortness D.=t~ g ~ 1 (t-\-f 2 . . . t + h 2 )^, we have in the three 
several cases ?=1, <P= respectively, 
H=l, 
<■= (1-f. 
y v= 
' AV 5 V 
H _ Vr+0 
V/ 2 ...+a 2 ’ 
^ ( 55 
„ )s v= 
” - ^ 
TT V/ 2 + 9. + S 
/ 2 ...+A 2 ’ 
f=^( „ 
„ )«, v= 
» - ai fcfW+t adt ’ 
MDCCCLXXY. 
5 H 
