292 THIRD REPORT — 1833. 



or in its applications to geometry. The terms tangent, co- 

 tangent, secant and cosecant, and versed sine, which denote 

 very simple functions of the sine and cosine, may be defined by 

 those functions and will be merely used when they enable us to 

 exhibit formulae involving sines and cosines, in a more simple 

 form. By adopting such a view of the meaning and origin of 

 the transcendental functions, the relations and properties of 

 which constitute the science of trigonometry, we are at once 

 freed from the necessity of considering those functions as lines 

 described in and about a circle, and as jointly dependent upon 

 the magnitude of the angles to which they correspond and of the 

 radius of the circle itself. It is this last element, which is thus 

 introduced, which is not merely superfluous, but calculated to 

 give erroneous views of the origin and constitution of trigono- 

 metrical formulae and greatly to embarrass all their applications. 



to the arcs of the equilateral hyperbola), whose formulse would bear a very 

 striking analogy to the formulae of trigonometry, properly so called. 



Abel, in the second volume of Crelle's Jotirnal, has laid the foundation, of 

 a species of elliptic trigonometry, (if such a term may be used,) in connexion 

 with a remarkable extension of the theory of elliptic integrals. If we denote 

 the elliptic integral of the first species 



X 



'^ ^/(l-c2sin2,^) 

 by 6, and replace sin \p by jc, we shall get 



-/ 



d X 



or more generally 



-X 



^{(l-a'2){l-t2a'2)} 

 d X 



V{(1 +e"*=) {l-<?x^)y 



If we now suppose a; = <p ^, V (1 — c^ «") =/ ^ and V (1 + e^-r") = F tf, it 

 may be demonstrated that 



<p{i) + e) - r+e2^2-^2-^-7^2-^» . 



, fd.f6'—c-(p6.(pd'.Y6. Fd' 

 f {0 + 6) — 1 + e2 c2 (p2 tf . (p2 tf' 



_ ¥d.¥6'-\-e^(p6.<pd' .fd.fd \ 



or if, for the sake of more distinct and immediate reference to these peculiar 

 transcendents, we denote 



(p S by sin 6 (elliptic sine of ff), 



f dhy cos 6 (elliptic cosine of ff), and 



F ^ by sur 6 (elliptic sursine of 6), 



