DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 349 
we find 
Assume Ap^=T{^-\-{a-\-hy, Aq^=-h^-\-{a—hy, . . . 
we shall then have the following' equations : — 
A-\-C=-Apq, 'B = {a-\-h){a~h) 
A-\-B={a-yp-q){a-\-q—p) ; C — B = {p-\-q-\-a){p + q-a) 
A={h-\rp-q){b-\-q—p) ; C=(jt?+9+&)(/)+y- &) 
ah = {p-\-q){p-q), 
Substituting these values in (129.) we obtain the resulting expressions 
p_ A[a + b){a-b)pq 
{p + q + b){p + q — b){a-^p-q){a-]rq-p) 
I • 
{a + b){a — b) {a-\-b){a—b) 
{aArp — q){a + q—py ''^~{p-\-q-\-b)\p-\-q — b). 
and if we denote by x. the criterion of sphericity, 
—b'^ / p + qA-a 'y / p + q — a 'y 
''^~a^[p + qy \p + q-{-b) \p + q — b) ^ 
(135.) 
(136.) 
. (137.) 
(138.) 
we may express the parameters and modulus of the elliptic integral of the third order 
and logarithmic form by a geometrical construction of remarkable simplicity when 
the intersecting surfaces are given, or when a, b, and k are given. 
Take BA=«, BD=i, and from O the point of Fig. 13. 
k 
bisection of AD, erect the perpendicular OC=^- 
Then (135.) gives j9=BG, ^=AC, and putting P 
and Q for the angles BACand ABC,a+^=2/> cosQ, 
a—b = 2q cos P. As p, q, b are the sides of the 
triangle BCD, and the angle BCD=P— Q, 
^n^2/ P-Q ^ {b+p+q){p+q-b) . 
H 2 ipq 
again as a, p, q are the sides of the triangle ABC, and. 
P + Q \ [a-\-p — q){a-Vq—p) 
2 ) Apq 
Substituting these values in (137.)? we get 
1*2 
cos P cos Q 
1 
cos 
rP+Q^ 
cos 
fP-Q\ 
' 
2’ 
^ 2 J 
{ 2 J 
cos P cos Gl 
cos P cos Gl 
m—- 
cos 
(139.) 
and if c be the eccentricity of the elliptic base of the cylinder, 
2 sin2P.sin2Gl 
sin2(P + Q) 
These are expressions remarkable for their simplicity. 
( 140 .) 
