DR, BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS, 355 
Stage of magnitude. But in the logarithmic ellipse the vertical cylinder will change 
its base with the change of the parameter. We shall see the importance of this 
remark presently. 
These ratios are — 
In the sphere 
In the paraboloid - 
j 
1 — m 
(172,) 
XLII. Resuming equation (157.) and developing it by a process similar to that 
applied to ( 127 .) 5 've get 
2 = 
ay 
s 
[1—2^ sin®(p]dip 
sin^<p 
Jcos^t’ 
k'^mn{\ — m) 
Now (151.) and (152.) give 
^=m, a.y=a^Ur, ^/y(«+(3)=j^, and 
Making these substitutions, we get 
^ [1 — i^sin^. p]cl(p y f dr 
Z = a^ ( 1 ~'^)J ["1 sin^cp ~ ^JcosV 
Now let rn=Q, then (165.) gives r=0, and we shall have 
2 = ajd(p\/ 1 sin^<p. 
(173.) 
(174.) 
This is the common expression for the rectification of a plane ellipse, whose greater 
semiaxis is a, and eccentricity i. This is case IV. of the Table, p. 316. 
We cannot arrive at this limiting expression by making e®=m=0 in (53.) ; for this 
supposition would render f=:0, which, throughout these investigations, is assumed to 
be invariable. 
XLIII. If, as in the case of the spherical parabola, we makew=m, orw=l— \/l— 
the values of | and | become infinite. What, then, is the meaning of the elliptic in- 
tegral of the logarithmic form of the third order, when n=m, or n —\~\/ 1 ? In 
the circular form of the third order, when m=n, n=i, and the spherical ellipse be- 
comes the spherical parabola, which, as we know, may be rectified by an elliptic 
integral of the first order. Not only do the ratios p ^ become infinite, but they be- 
1 —71 • 
come equal, for ^=^17^=1? when m—n. What, then, does the integral in this case 
signify? It does not become imaginary or change its species. 
Resuming the equation established in (133.), 
2\2n — ?—rfi'\ 2 
V — n)[? — n) k 
n^n-\ 
