439 
FOEM OF TANGENTIAL EQUATION. 
(a cos <p— sin <p— gf— Jc 2 ; (300) 
gVfl ab{(a cos<p — /) 2 + (6 sin <p — g) 2 — kft)d<p 
% § g{a 2 sin 2 p-}-6 2 cos 2 <p} 
abdip ab\a 2 + b 2 +f 2 +ff‘ 2 — Jc' z — 2af cos <p — 2 bg sin <p}d<p 
q g{a 2 sin 2 <p + b 2 cos 2 <p} 
Put sincp= 1 ^-g; then cos <p=|^ 2 , and Making these substitutions, we 
2 abdz 2ab { (a 2 + 6 2 +/ 2 +^ 2 — A 2 )(l + ,g 2 ) 2 — 2a/(l — .g 4 ) - 4bgz ( 1 + -z 2 ) 1 dz 
VZ 
4a 2 g 2 + 6 2 (l-.g 2 )} VZ 
, • (301) 
where Z stands for the quartic 
{a(\-z^)-fO-+^)Y+\2tz-g(\+z:‘)Y-lc‘(\+zJ . . . . (302) 
144. In order to reduce still further the expression (301), we must decompose 
9 z {(a* + b*+f*+f-W(l+i%)*-2qf(l-**)-4bgz(l+z*)} 
4 «V + 6 2 ( 1-^ 2 ) 2 
into simpler fractions, or say the fraction <J> into simpler fractions. 
Let us, for shortness, put f for the expression a? -\-g' 2 — k 2 . Making this 
substitution, and denoting the eccentricity of the conic -^-{-^—1 by e, we find, by 
dividing &c., 
-_2 a . Q j,. 2ab{4b 3 gz 3 + 4a‘ i {e' 2 t‘ 2 + c<f(l+e2)}z‘ i + 4b 3 gz + b‘ 2 i.‘ 2 4-4ab'*f\ 
b Ui ~ Aa j) {^ 2 + a 2 (I+e) 2 }{ 6 2 2 2 + a 2 (l-e) 2 } ' 
Then decomposing the fraction still remaining, and substituting, we get, after some 
reduction, 
41ft g /' zdz 4 b 2 g zdz 
«J { (1 + e)z* + (1 — e ) } VZ T J { (1 e)^ 2 + (1 +e)} VZ 
b{8afe+{l-e){l+3e)t*} P dz 
2ae(l + e) J { (1 +e)z' 2 -\r (1 — e ) } VZ 
b{8afe-(\ + e){\-3e)f} f dz 
2ae(l — e) J { (1 — e)z 2 + (1 — e ) } VZ * * * ‘ ^ ' 
Since Z is a quartic function of the variable, each of these integrals belongs to the 
domain of the elliptic integrals. (See Cayley’s £ Elliptic Functions.’) 
MDCCCLXXVII. 3 Q 
