DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 387 
Since 
Again, as 
and 
it follows that 
sec(<p-‘-<p)=secY+tan^(p, and tan((p-i-^)=2tan(p sec^, 
sec(^-J-<?)) + tan((pA-(p) = (sec® + tan(p)^ 
sec((p-‘-9-L-(p)=sec(^-*-<p) sec<p-\-ta.n{(p-i-(p) tamp, 
tan((p-i-(p-i-(p) = tan(p-t.^) secp+sec(p-i-p) tan<p, 
sec (p p -1- (p) _j_ tan (<p -t- p -1- pi) = (seep + tamp)®, 
and so on to any number of angles. Hence 
sec(p-J-p-J-p... to /?p)+ tan(p-J-p-i-p to wp) = (secp+tanp)“. . . , (342.) 
Introduce into the last expression the imaginary transformation, tanp=\/ — 1 sinp, 
and we get Demoivre’s imaginary theorem for the circle, 
coswp+\/ —1 sinwp=(cosp+\/— 1 sinp)”. 
Let a be conjugate to and aj, while a, as before, is conjugate to p and x- Then 
we shall have tan^=tan(p-^%-*--4/), or 
tan [<p X 'll) — tanp sec;^ sec^/ + tan^ sec^/ seep + tan^' seep sec;)^ + tanp tanp^ tan^/, . . . (a-.) 
see(p 4/) = seep see;^ see\I/ + seep tan;^ tan\I/ + see;:(^tan4/ tanp + sec4/ tanp tan;;^, . . . {p.) 
aDd sin(* a- V - 4.) ^ sinj^ + sinx + sin4, + ain^sinxsin4, 
1 +sin;)^ siny + siny sinp + sinp sin;^ 
whence, in the trigonometry of the circle, 
sin(p +;;^ + 4/) = sinp cos;;^ eos4' + sin;);^ eos^/ eosp + sin\I/ eosp eos;;(; — sinp sin;^ sinvj/, 
cos(p +x + 4') = cosp eos;)(^ eos4/ — eosp sin;^ sin4' — eos% sin4' sinp — cos4' sinp sin;^, 
tanp 4 tanx 4 tan4/ — tanp tan;^ tan4' 
tan(p4x44^)^ 
((X.) 
(P-) 
(r.) 
(s.) 
1 —tan;)(^ tan4^—tan4/ tanp — tanp tanj,^ 
LIX. Let (k.a), (Ji-x) denote three parabolic arcs measured from the vertex 
of the parabola whose parameter is k. 
The normal angles of these arcs are a, p, and %; a;, p and being conjugate ampli- 
tudes. Then 
2(A-.p) = /: tan p seep-H/fj^. 2{k.x) = ktanx 2{k.a}) = k tan^y seca;+^(^ 
whence, since because a, p, and x conjugate amplitudes, 
{k.a)) — {k.(p) — {k.x)=ki(inui3.n<p tan^; (343.) 
Let 3/, y, y' be the ordinates of the arcs (k.cp), (k.x), and (k.ai). Then ?/ = A-tanp, 
y' — k ianx, y'' = ktiinci), and the last expression becomes 
{k.a,)-(k.p)-{k.x)=^f (344.) 
If we call an arc measured from the vertex of a parabola an apsidal arc, to 
distinguish it from an arc taken anywhere along the parabola, the preceding theorem 
