DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 3l 1 
We may, using successively the equation determine in terms of j the 
successive values of i, i^, and of^’p &c., as follows : — 
;_iz4 ; -[!-/]' / _r (i+#-2V* r _r i+/-2*2^(^+.;)¥ in 
‘ l+J’ “ Ll+;iJ ’ Lfl+;)i4-2hTj ’ Li+;i + 2^2i(l+;)h'iJ 
. ^ r (l +»* + 2^/-2*2^2*(l +/)*(! 
’ L(1 +;-)* + 2h'^ + 2i242i(l +;■*)(! + ;)W J 
-(1 + f )^ + 2 y '* + 2 ^ 2 * 2 i{l +J®)(1 + j )^ j ' 
Hence we may derive the successive values j p j i,, j m in terms of j. 
J 
For 
j: 
■2_ 2y 
a+jV 
j: 
222/(1 +i) 
( 1 +/) 
j: 
22212^/(1+7/(1+/)^ 
[(l+7/ + 2i/]' 
•2 _ 22212*2^(1 +/) ( 1 +7// [(1 + 7/ + 2*7*]2 
r/, , . ^ 4 ^ 4 -/, , •^ 4 •i ^4 ’ 
P 
(105.) 
(106.) 
[(1+/) + 2*21(1+7/7/^ 
(22212*212*) (1 +7*)*(1 + 7/7-^[(l + 7/ + 2*/] [(!+/) + 2*2/1 +7/7?]^ 
[(1 +7/+ 2*7! + 2*2l2*(l +7*)*(1 +/*ii^]‘^ 
We may express the coefficient Q, or the continued product (>^j,jpj,pjm, &c., in 
terms of j, the complement of the original modulus. Including in our approximation 
the fifth derived modulus, we get 
(2)1. (2)^'!'*. (2)^"**‘ll . (2)^ *“*“11*"*. (2)^l"*'*l*“**”^(77*///i*) 
Q 
'(1 + 7')*(l +/)*[(! + 7/ + 2*7 t]*[( 1 +7*) + 2*2/1 +;/7*][(l +7/+ 2*7! + 2*2l2*(l +7*)/l +i)/‘'*]2 
XXX. It may not be out of place here to show, although the investigation more 
properly belongs to another part of the subject, that the arc of a spherical parabola 
may be represented as the sum of two elliptic integrals of the third order, having 
imaginary parameters ; or in other words, that every elliptic integral of the Jirst 
order may be exhibited as the sum of two elliptic integrals of the third order, having 
imaginary reciprocal parameters. 
Assume the expression given in ( 58 .) for an arc of the spherical parabola, the focus 
being the pole, and g the angle which the perpendicular arc from the focus, on the 
tangent arc of a great circle to the curve, makes with the principal transverse arc, 
du, , "If sinytana- 
-7 -- ^ ^ +tan 1-7= . g ^ 
V 1 — cos"'y sm'^jx [ v 1 — cos"^y sin^/x. 
Let cosy=i, siny^/, and to preserve uniformity in the notation, write <p for f/j. 
Then differentiating the preceding equation, it becomes after some reductions, 
de- 7 [1 — sin^fp + cos^ip +/ sin^tp] 
dip [cos^ifs — sin2(p cos^^p +7'^ sin2(p] \/l — 
Now the numerator is equivalent to 2/1 — ^^sinY), and the denominator may be 
written in the form 1 — sin^ip+«^ sin^- But 2^= ^^(^2 -[-/), hence this last expression 
maybe put under the form 1 — 2/)^ sin'^+ji sinY + // sin^ip. expression is the 
sum of two squares. Resolving this sum into its constituent factors, we get 
dcr 27 ( 1 — sin^ip) 
dip [^1 _j(f_y y' — l)sin®ip] V' 1 — sin^® " 
(b.) 
(IO7.) 
