ON THE TRIGONOMETRY OP THE PARABOLA. 69 



Adding these expressions together, and introducing the relation established 

 in (2), we find 



dig _ d<^ I <^X (5) 



cos W COS (p COS X 



This is the differential equation which connects the amplitudes w, ^, and x- 

 As w, ^, and x are supposed to vanish together, we shall have by integration, 



r_rf<u_^r_rf^ r_^. ^5^ 



J cos (1) J cos ^ J cos X 

 or in the more compact notation, 



(* sec w(?w=lsec0c?^ + i'secx«?X • (^)* 



Hence if w, ^, and x are connected by the relation assumed in (1), we shall 

 have the simple relation between the integrals expressed in (5). 



II. If in (1) we make the following imaginary substitutions, that is to 

 say, put '^— Isina for tan^, '^^— lsin/3 for tanx. 'V— Isiny for tan w, 

 cos a for sec 9, cos /? for sec X) cos y for sec w, and change -^ into + and -r 

 into — , we shall have sin y= sin(a+/3) = sinacos/D+ sin /3 cos a, the well- 

 known expression for the sine of the sura of two arcs of a circle. 



We shall show presently that an arc of a parabola measured from the 



vertex may be expressed by the integral j*sec0(/y, Q being the angle which the 



normal to the arc at its other extremity makes with the axis, or the angle 

 between the normals drawn to the arc at it? extremities. 



-■- 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, the following expi-essions in parallel columns are given ; premising 

 that the formulae marked by corresponding letters may be derived singly, 

 one from the other, by the help of the preceding imaginary transformations. 



In applying the imaginary transformations, or while tan is changed into 

 V— 1 sin tp, sec into cos (p, and cot^ into — V—\ cosec 0, -^ must be 

 changed into +, and -r into — ; as also \sec(j)df into ^-v — If. 



The reader who has not proceeded beyond the elements of trigonometry 

 may assume the fundamental formula as proved. He will find little else that 

 requires more than a knowledge of plane trigonometry. 



* The relation between the conjugate amplitudes lo, <p, and x, was originally obtained in 

 this way. In the theory of elliptic integrals, any three conjugate amplitudes are connected 

 by the equation 



cosa) = cos0cos X— sin0 sinx 'V 1 — i^sin- w 



» is called the modulus. When we make i=0, we get 



cos w = cos cos X — sin sin % ^'^ '^= 0+X i^^ the trigonometry of 

 the circle. When we take the complement of 0, or make i = l, we get 



secw= sec ^ sec x+ tan tanx '^^ '^ = <p-^x 

 in the trigonometry of the parabola. Whence, as above, 



tan oj = tan sec x+ tan x sec 0. 

 •f- I hardly need to remind the advanced reader, that this is the imaginary transformation 

 by which we are enabled, in elliptic functions of the third order, to pass from the circular 

 form to the logarithmic form, or to pass from the properties of a curve described on the sur- 

 face of a sphere to its analogue described on the surface of a paraboloid of revolution. See 

 the author's paper " On the Geometrical Properties of Elliptic Integrals," in the Philosphical 

 Transactions for 1852, pp. 362, 368, and for 1854, p. 53. 



