404 UR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Let, as in (25.), tanX^= coss tan(p=^^ tan<p. Substituting, we get the expression 
sin? sin»} sincp coS(p 
•y/ { 1 — sin^s sin^f) ( 1 — sin^>j sin^ip) (390.) 
In supplemental spherical ellipses, since sin?? and sins* are respectively equal to 
sins' and sinV, we infer, therefore, that in supplemental spherical ellipses the inclina- 
tions to the plane of xy of the tangents to the curves are the same, when the ampli- 
tudes <p are the same. 
dv 
If we now differentiate this expression, and make ^=0, we shall find that 
tan-®=^^. If we substitute this value of tanip in (390.), we shall get 
tant'=tan (a— /3), or v=a — j3 (391.) 
Hence the maximum inclination to the plane of xy of the tangent to the spherical 
ellipse is equal to the difference between the principal semiarcs. It is remarkable 
that the point of the curve which gives the maximum difference between the arcs, 
which together constitute the quadrant of the spherical ellipse, is not the point of 
greatest inclination. For this point is found by making tanV=*^ ; while the point 
of maximum difference is obtained by putting tan^<p=^j^^. This is the more worthv 
of notice, as we shall find the two points — the point of maximum division, and the 
point of greatest inclination — to coincide in the logarithmic ellipse. 
If we take the two plane ellipses which are the projections of the spherical ellipse, 
one being tbe perspective, and the other the orthogonal projection, and seek on 
these plane ellipses their points of maximum division, we shall find that the angles, 
which the perpendiculars on the tangents, through these points of maximum division 
of those plane curves, make with the principal arc, are the values which must be 
assigned to the amplitude (p, to determine the point where the tangent to the curve 
has the greatest inclination to the plane of xy, and the point which divides the 
quadrant into two parts, such that their difference shall be a maximum. This is 
])lain ; for the semiaxes of one ellipse are /r tana, tan/3 ; while the semiaxes of the 
other are k sina and k sin/3. And these angles are given by the equations 
, tana , „ sina 
tamX=:; — n; and tan%=^-^- 
tanp ’ ‘ sinp 
On the Maxhnum Protangent in the Logarithmic Ellipse. 
LXXV. If we follow the steps previously indicated, and differentiate tbe expression 
found in (165), 
sinr 
mil sinip cosp 
y/l — sin^p ’ 
. (a.) 
* Theory of Ellij)tic Integrals, p. 19. 
