709 



If we pass from the circle to the ellipse, the addition of the arcs 

 becomes more complicated, the product of the sines being multiplied 

 by a radical. This relation was first obtained by Euler as the inte- 

 gral of a differential equation. I have called it elsewhere, for con- 

 venience, the elliptic equation between three amplitudes. Moreover, 

 we are no longer able to take the simple sum of the arcs, but we have 

 to add in an algebraic quantity, which is a multiple of the products 

 of the sines of the three amplitudes. 



The comparison of hyperbolic arcs hitherto has been matter of 

 still greater complexity, as it has been usually handled simply by 

 reducing each hyperbolic arc to two elliptic arcs. The complexity, 

 of course, is thus doubled. 



It seemed to me, however, that the ellipse is so completely the 

 analogue of the hyperbola, and that it is so easy to pass from one to 

 the other by an imaginary transformation, that a similar analogy 

 ought to pervade the comparison of their arcs. Hence that there 

 must exist some formula for the comparison of hyperbolic arcs, as 

 simple as that for elliptic arcs. 



A slight modification of Jacobi's second theorem has enabled me to 

 find the analogue which I required. 



If we take Euler' s elliptic equation, and substitute the following 

 changes, 



cosine of amplitude into secant of amplitude, 

 sine of amplitude into tangent amplitude x V' 1, 

 sine of modulus into cosine of modulus, 



we leave the equation entirely unaltered, except in form. Even this 

 form may be obtained directly by a simple transformation, of purely 

 algebraic character. Hence it follows that all the consequences of 

 this imaginary transformation are allowable. 



If we apply these transformations to the elliptic integral of the 

 first kind, we have Jacobi's second theorem. 



If we apply them to the integral of the second kind, which repre- 

 sents the elliptic arc, we pass, after an obvious reduction, to the arc 

 of the hyperbola. The algebraic addition to the sum of the arcs is 

 simply changed from a product of sines to a product of tangents. 



Other considerations enable us to verify the theorem, when once 

 it is obtained. These verifications are merely algebraic, and would 

 scarcely be intelligible if read aloud. 



