The collected mathematical papers of Arthur Cayley.

603] ON THE POTENTIAL OF THE ELLIPSE AND THE CIRCLE. 297
We thus have
1 dA 1 f2sin2q afcos q-a2cos2
c d - DA ^ + HIA
whence, by integration,
1 tan-' c cotA do + (afcos f -a2 cos2 ) d,f sinS d
c 2+cc2taf cos A HA fJ A
which is the required formula of transformation.
28. Multiplying by c2, and subtracting from the value of V, we find
V=C tan-' ( ot — bz ~ (+af(in
a2 + c2 -afcos a A ) f
+ f2 +/2- af c2 c) cos (/-acos b)do
A j (c2 + a2 sin2 <,) A
which is to be taken between the limits 0 and 2wr; viz. we thus have
V = -2cr + 2 f (C2 +~2- - ayeos b) dob 02a cos 0b (f - a cos c ) db
o A o (c2 + a2 sin2 ) A
agreeing with a former result.
29. But this former result, previous to the final step of taking the integrals
between the limits, was
V= 2asin < log( f-cos )+A - c tan- )an
-a cos + V/a2 + c2 c
+ (2 +f2 - af cos ) d fcos (/f-a cos ) d.
A J (c2 + a~ sin2 ) A;
viz. the integrals are the same, but the integrated terms are altogether different;
the explanation of course is that the b's are different in the two formula, which therefore do not correspond element by element but only in their ultimate value between
the limits.
30. In order to discuss numerically the Potential of the Circle,
/ (t - 0. t + )dt
fj t (t +f2) t +/2 '
this must be reduced to elliptic functions. Writing t=0 +2, we have
V= 4f 2 X:V +I3'dx
( + ) (Ix + ( 2 )3'
C. IX. 38
38