386 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
The preceding formula will become, on substituting these values, and dividing by a/ — 1, 
sinai' =&in<p' cos%' + sin%' cos<p', 
the well-known trigonometrical expression for the sine of the sum of two circular arcs. 
Hence, by the aid of imaginary transformations, we may interchangeably permute the 
formulae of the trigonometry of the circle with those of the trigonometry of the parabola. 
In the trigonometry of the circle, and in the trigonometry of the parabola is 
such a function of the angles and as will render tan [( 9 ,%)] =tan?5 sec/.-f tam^ sec^. 
We must adopt some appropriate notation to represent this function. Let the func- 
tion be written so that _tan(?J-i-x) = tan?) sec^^-f tan):: sec®. This must 
be taken as the dejinition of the function 
In like manner, we may represent by tan(^-r%) the function tan?) sec^— tan^ec?. 
In applying the imaginary transformations, or while tan?> is changed into^- 1 sin?, 
sec? into cos?, and cot? into cosec?, -i- must be changed into -f and into-. 
-J- and -r may be called logarithmic plus and minus. As examples of the analogy 
which exists between the trigonometry of the parabola and that of the circle, we gi^e 
the following expressions in parallel columns ; premising that the formulae, marked 
by corresponding letters, may be derived singly, one from the other, by the help of 
the preceding imaginary transformations. 
Trigonometry oj the Parabola. 
tan(?->-):)=tan? sec)(:-l-tan)'sec?. . . . («•) 
tan(?-r%) = taB?’ secx— tan)c sec?. . . . (|3.) 
sec(?:^%)=sec? sec)(:+^3'ii‘P 
• 1 . sin? + sinx _ 
Sin(? x) 1 + sin? sin;);: 
sin?-sinx / \ 
sin(?-r%) = rz:^hi^^ 
Let (p=x- 
tan(?-L?) = 2tan? sec? 
sec(?-L?) =sec^?-!- tan^? (^0 
s>n(?^?) = l+sih^ Vd 
/'iL /'dl 
pycosip I 0 ycos?) qJ cosip — qJcosP 
&ec<p=- ^ ^ tan?= 3 ^ 
— 1 tan (?-*-?) = (sec? 1 tan?) • 
sec(?-L?)-l ( \ 
tan^?= 2 
Let the amplitudes be <p-^x ^^^d ?-rX- 
tan(?->-)(:)tan(?-r%) = tan^?’— 
Trigonometry of the Circle. (341 . 1 
sin(?-l->:) = sin?cos)(:-l-sin)::cos?. . • . (a.) 
sin(?— )(:) = sin?cos):— sinxicos?. . . ■ (b.i 
cos(?±)(:)=cos?cos%+sin?sin):. . . . (c.) 
tan? + tanx 
tan(? + X:) = i_tan?'t^ 
tan?-tanx CgN 
tan(? x) i-ftan?tanx: ' 
Let (p=X- 
sin2?=:2sin?cos? 
cos2?=cos^?— sin®? (bid 
tan2f>=i:rtj„5^ 
cos?= 2 ’ sin?= 
1 -l-sin2?=(cos?-l-sin?)® (^■ 
. „ 1 — cos2? /px ' 
sin®?= 2^ ^ 
Let the ainplitndes be ?-|->: and ?— %• 
sin(?-f%)sin(?-x)=sm®?-sin®):. . . . (d- 
