312 
PKOFESSOR G. H. DARWIX OX THE PEAR-SHAPED FIGURE 
In accordance with the usage in elliptic integrals, I write 
= 1 — K“ sin^ iff 
under the integral sign, or 1 — K^sin^y outside the integral 
I shall also for brevity write 
A]^ = 1 — (7' sin~ xfj 
under the integral, or I — q~ sin~ y outside the integral. 
We have then 
{Po) (g-)) = (^o)]' • 
1 apply this formula successively to the several functions, as given in (9) to (17), 
and introduce the abridged iiotation just defined, hut T do not reiterate the special 
meanings to he attached to tlie symbol <[ in each case. 
Since Po (^) — 1’ have (dropping tlie now unnecessary suffix 0), 
sin- y 1 ^ 
p/ (^) Q/ = K cot' y [' dyjj, 
(c) (Q^l(j/) = 
K A- n sin- -v/r 
iQj (i') and ia/(a) ©/ (:.) = " I tAt <'4, 
Sin >y j Q jij 
P.‘ (a) Qd (a) = " f’ -"'Af- J4,, 
^ \ \ / .suP y JijCOS'-v/rA ^ 
^^ 2 ^ (c) (Q.d (v) = ~~ j 
' ^ ~ ' Slip7 Jq A’ ^ 
^ COS- yA- [y sill^ ^jr j 
P-r (O Q-=- (0 = fG, 
ja, (a) ©ae and 5 P 3 =(a) © 3 ^ (a) = P'- t 5^-^ 
sill 7 •' 0 “^1 
P3>(a) Q,ae and P 3 =(a) Q, 3 („) = f’ 
103'(a) ffi3'(a)and ft 3 (a) ©33(a) d^, 
bill 'y j Q ^ 
r-k o / \ o zc COS- vA- (y sill® ylr 
P3'0^)€U= : c ■ -T^V.dxl, 
sin® 7 Jcos- -v/rA ’ ■ 
(25). 
