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XXXI. On a particular case of Elliptic Integrals whose Para- 

 meters are Imaginary. By the Rev. James Booth, LL.D., 

 F.R.S. 6fc.* 



THE arc of a spherical parabola may be represented by the 

 sum of two elliptic integrals of the third order, whose 

 parameters are imaginary and reciprocal. The spherical conic 

 section, whose principal arcs 2« and 2/3 are connected by the 

 equation tan^a— tan^/3=l, is called the spherical parabola. 



If we assume the expression given at page 32 of the Theory 

 of Elliptic Integrals t, for an arc of the spherical parabola, the 

 focus being the pole, and (j) the angle which the perpendicular 

 arc from the focus, on the tangent arc of a great circle to the 

 curve, makes with the principal transverse arc, 7 being the modular 

 angle, or the angle between the focus and the adjacent vertex, 



.=sin7/^, ^"^ . +tan-'r /^''T'^ . T (1) 

 ^ V 1 — cos^7sin^^ L v 1 — cos^7sm^^-' 



Let cos 7=1, sin 7=;'. Then differentiating the preceding 

 equation, it will become, after some reductions, 



dar ^' [1 — i^ sin^ (f> + cos^ <f) +j'^ sin^ 0] ,_ , 



d^ ~ [cos^ (^ - i^ sin^ ^ cos^ <^ +fsin^(f)'] \/T^^^~si^' ' ^ 



Now the numerator is equivalent to 2^(1— i^sih^0), and the 

 denominator may be written in the form 1 — 2i^ sin^ ^ + «^ sin''^. 

 But i-=:i^{i^+j^), hence this last expression may be put under 

 the form 1— 2i^sin^^ + «^sin'*^ + z^^'^sin^</). This expression 

 is the sum of two squares. Resolving this sum into its consti- 

 tuent factors, we shalJ find 



^_ 2;'(l-z'^sin^0) 



^<l> [l-i{i+j) \/^l)sm^(j>] li-i{i-j ^/-l)sin20] \/l^hin^(f)' 



This product may be resolved into the sum of two terms. Let 



A B 



■ + 



(3) 



(5) 



;i-i(t+7V-l)sin2</)] -/l-i^siu^^^ ll-i{i-j\^-l)sin^(}>'] x/l-ihin^cfy' 

 or reducing these expressions to a common denominator, 

 d^_ (A + B) - (A + By sin^ + (A-B) \^^lij sin^ (f> 

 d<f> [1 -i{i -rj \^^T)sm^(}>'] ll-i{i-j V ^)sin2</)] a/] -i^sin^,^" 



* Communicated Ijy the Author. 



t The Theory of Klhptic Integrals and the properties of Surfaces of the 

 second order ajiplicd to the investigation of the motion of a hody round a 

 fixed point. 8vo. London, G. Bell. 1851. 



