416 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Multiply the first of these equations by 2 and add them, 2 will be eliminated. In this 
way we may successively eliminate 2^, 2„, 2,,^, until ultimately we shall have 
22v 2 — — 
p being the number of operations, and denoting by F and E, the sum of the integrals 
of the first and second orders, by "F the sum of the right lines, and by 11 the sum of 
the parabolic arcs. 
If in (401.) and (416.) we substitute the coefficients of the derived integrals as 
transformed in (404.) and (430,), the relation between the original and the derived 
integrals of the third order will be, 
/ ^+” \ /— r ^ I /— r dip A+M / — C— _A4 4 
\ n ) ^ j(l-)-»isin^ip) Vl— i^sin^p \ »* / j(l — msin^p) Vl— i^sin-ip \ n, ) siii-.p)A/l — i/sin-ip’ 
for the circular form or the spherical ellipse, and 
, — r dip 
\ m } ^ j(l — m sin^p) V^l — sin*p 
for the logarithmic form, or the logarithmic ellipse. 
LXXXIII. There are several plane curves, whose lengths we may express by elliptic 
integrals of the third order. For example, the length of the elliptic lemniscate, or the 
locus of the intersections of central perpendiculars on tangents to an ellipse, is equal 
to that of a spherical ellipse, which is supplemental to itself, or the sum of whose prin- 
cipal arcs is equal to t. We cannot represent elliptic integrals of the third order 
generally, by the arcs of curves, whose equations in their simplest forms contain only 
two constants. Thus let a and b be the constants. We shall have two equations 
between the constants the parameter and the modulus of the function, i=f{a, h). 
n=f'{a, b). Assume a as invariable, and eliminate b, we shall have one resulting 
equation between i, w, and a, or F(a, i, n)=Q ; or n depends on i. 
When there are three independent constants, as in the preceding investigations, 
a, b, and h, we shall have i=f[a, b, h), n=f{a, b, k). Eliminating successively b 
and k, we shall have tv/o resulting equations, instead of one, F(a, k, i, w)=0, and 
F'(a, b, i, n)=0, or i and n depend on two equations, and may therefore be inde- 
pendent. 
-(V) 
sin^p) sin^p 
di/- 
Vi- 
!435 
1/ S'll-vP 
ERRATA. 
Page 319, last line, dele n. 
320, line 5, for page 6 read page 316. 
328, line 12, /or (47.) read (46.). 
329, line 15, /or (32.) read (31.). 
1-y 
331, line 7 from bottom, /or n-=.m=-i, read n=m=-^rT'‘ 
338, to the last line add, i being here the eccentricity of the base of the elliptic cylinder, 
372, line 11, /or Case XII. read Case XIII. 
389, line 14, /or Bernouilli read BERNOLani. 
