DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 337 
Introducing this relation into the last formula, and equating together the equivalent 
expressions for the arcs in (84.; and (85.), we get for the resulting equation, 
+ (l+i) tan 
fi-i' 
(i +j> 
1 sin\J/ cosvj/ 
'1-7) 
.i+jJ 
( 86 .) 
We shall now proceed to show that the common formula for the comparison of 
elliptic integrals having the same modulus and amplitude but reciprocal parameters, 
is, in this particular case, identical with the geometrical theorem just established. 
The formula is, in the ordinary notation, 
2^.(c,^^.) = F,(■4.). 
1+c L 
1 1+7 
We must accordingly show that, c being tan% and therefore T+c ='2 
(87.) 
(l+i)tan 
CGSv}/ 
+ (l+i)tan 
> . 
(1 +tan^s) tan\J/ 
sin®4' 
( 88 .) 
If we write r, r' and 0 for these angles respectively, we have to show that 
0=2(r+r') (89.) 
T+r' is the arc of the great circle, which touches the spherical parabola, intercepted 
between the perpendicular arcs let fall from the centre and focus upon it. 
We must, in the first place, by the help of Lagrange’s equation between the ampli- 
tudes, established on geometrical principles in XXIV., reduce these angles to a single 
variable. {Jj is taken as the independent variable instead of as the trigonometrical 
function of in terms of f/j is in the first power only. 
We have, therefore, 
tan0= 
2 tan\l/ 
tanr = 
( 1 + 7 ) 
sin4/ cosvl/ 
tanr' = 
j tan/ji. 
sin^/x 
2x2 
( 90 .) 
