340 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Nowj;=^, and as in (98.) and (74.), 
whence — (99.) 
Thus a simple ratio exists between the arcs, diminished by the protangents, of two 
consecutive confocal spherical parabolas. 
TT 
When the functions are complete, ^ is taken between 0 and ^ ; <p therefore, as in 
article XXIV., must be taken between 0 and or ; but when the amplitude is taken 
between 0 and or the function is doubled. Moreover, when the functions are com- 
plete, the point Q coincides with B ; so that in this case the complete function 
represents, not one, but two quadrants of the spherical parabola, the focus being the 
pole. Hence as 7-=^, r'=or. 
Whence putting C, C', C", C'", &c. for the circumferences of the successive confocal 
spherical parabolas, derived by the preceding law, we may write 
C -or = x/j (C^ -or)' 
C, -or=x/^(C,, -or) 
C„-or=Vj;,(C^„-or) > 
C/;y — = — ^) 
Cjy or ^ 
( 100 .) 
Multiplying successively by the square roots ofj &c., adding and stopping 
at the fifth derived parabola. 
&CC. (C,-or). 
Let this coefficient be \/ Q, and we shall have C — or=\/Q(Cv— “s”). . . . (101.) 
Now we may extend this series, until the last of the derived spherical parabolas 
shall differ as little as we please from a great circle of the sphere. Let the circum- 
ference of this last derived spherical parabola be C,. Then C»=2‘r, and (101.) becomes 
C = or(l-f>/Q') (102.) 
Hence calculating the quantity Q', we may express the circumference of a spherical 
parabola by the circumference of a circle. 
When all the spherical parabolas are nearly great circles of the sphere, 
nearly ; and I, nearly. Whence Q'= 1, nearly ; or 
C = 2or (103.) 
When the spherical parabolas are indefinitely diminished, 
= nearly, and therefore Q'=0 nearly ; 
or (104.) 
Hence the circumferences of all spherical parabolas lie between two and four 
quadrants of a great circle of the sphere. 
XXIX. Denoting the angles at the centre of the sphere, subtended by the halves 
of the semiparameters of the applied confocal parabolas, by y, y', y", &c., we have 
cosy=i, cosy'=y, cosy"=^y, cosy"'=q,„ and siny=J, siny'=:jp siny"=jy, sin/'=y„,. 
