DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS, 323 
of the curve AQB in a right line ZP, or p, which passes through the pole, the point 
of contact of (0) with the sphere. This line p being in (IT) must be perpendicular 
to t. The plane (II) will also cut the sphere in an arc of a great circle Zns per- 
pendicular to Knx, the tangent arc to the spherical curve ; for these arcs must be at 
right angles to each other, since the planes in which they lie, (11) and (T), are at 
right angles. Let P be the distance OP of the point, in which the plane (IT) cuts the 
right line t, from the centre of the sphere ; r the distance ZQ of the pole of the plane 
curve to the point in which t touches it, r being the angle which t subtends at the 
centre of the sphere, and k its radius, 
j9 = ArsinOT, ^=P tanr J ^ ' 
T is the angle between OQ and OP. 
Let d^ be the element of an arc of the plane curve between any two consecutive 
positions of R, indefinitely near to each other ; hda the corresponding element of the 
spherical curve between the same consecutive positions of R, Then the areas of the 
elementary triangles on the surface of the cone, between these consecutive positions 
of R, having their vertices at the centre of the sphere, and for bases the elements of 
the arcs of the plane and spherical curves respectively, are as their bases multiplied 
by their altitudes. Let S and S' be these areas ; then 
(a.) 
But the areas of triangles are also as the products of their sides into the sines of the 
contained angles, i. e. in this case as the squares of the sides, or 
S ; S' : ; R^ : (b.) 
do- P ds 
dX~R2 dx 
putting for d.? its value given in (31.), 
d<r P/d^;, ] ,,, 
dA~R2{dx2+/^| 
Now /)=PsinOT, P^=R^— and P^=F+p^; 
whence and 
Substituting these values in (d.), 
dx— sintv-t-j^ 2 |r ^^2 dx dxj te.j 
We now proceed to show that the last term of this equation is the differential of 
the arc, with respect to subtended at the centre of the sphere, 
t P 
This arc being r, tanT=p, cosr=^- 
