DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 413 
To determine the axes of the base of the cylinder, whose intersection with the 
paraboloid gives the derived logarithmic ellipse. 
Since 
mp ^ — infer from (1/1 0? 
we shall have, substituting the preceding values of and w,, 
a - {a + bfk^ b^^_Aab[k^+{a-bY] . 
k^ \k^-AabV’ F [k^-AabY ’ ' \a + bj 
- + {a + hf 
k{a-bfA 
(419.) 
Ik^-AabY [k^-AabY 
When A-=oo, or when the paraboloid is a plane, a^={a-\-b), b=2\/ ab, which are 
the values of the semiaxes of a plane ellipse, whose eccentricity is 
as we should have anticipated, for these are the values found in LXXVII. and LXXIX. 
for the axes of the derived plane ellipse. 
When 
, . mn /I— i\^ , 
m = / 2 =l-j, =r, and 
Hence, when the original logarithmic ellipse is of the circular form, the first derived 
ellipse is a plane ellipse. 
When lf=Aab, (418.) shows that m=n^, or j=j= oo, as in XLIII. ; but is 
equivalent to n=m(\/ l+J+vOy. 
Whenever therefore this relation exists between the parameters and modulus of 
the original integral, the first derived integral will represent the circular logarithmic 
ellipse, which may be integrated by functions of the first and second orders. Accord- 
ingly whenever the above relation exists between the parameters, the integral of the 
third order may be reduced to others of the first and second orders. 
If in the second, third, or any otlier of the derived logarithmic ellipses, we can 
make the parameters equal, this derived ellipse will be of the circular form, and its 
rectification may be effected by integrals of the first and second orders only ; accord- 
ingly the rectification of all the ellipses which precede it in the scale, may be 
effected by integrals of the first and second orders only. 
We may repeat the remark made in LXXIX. The derived functions of two 
integrals of the logarithtnic form with reciprocal parameters, have themselves reci- 
procal parameters. 
LXXXII. If we now add together (162.) and (163.), we shall have 
4(« — m'' 
7 
'df 
Vi 
dr 
(420.) 
We must now reduce this equation into functions of -4/ instead o? (p ; 4' 9 being 
connected, as before, by the fundamental equation 
U\.n{4 — P)=J tamp. 
