ME. 0. W. mupptwet.T) on the OOMPAEISON OP HTPEEBOLIC AECS. 175 
One lesson we may learn from this process is, that the proper expression for the 
negative parameter greater than unity is -(1 + cos’ « tan’ »). In geometrical researches 
this remark will probably lead to simplicity. Leoendee has deliberately avoided the 
discussion of this form of the parameter*. His reason was, that the complete integral 
presents itself in the form of co — co . , i o j 1 1 ^ 
The tabulation of the function Yu would only require a table of double entry . 
It may be as well to notice that the equations {a.\ {b.\ (c.) are solved by auxiliary arcs 
as follows : 
Assume ^ a 
tan >J2~ <pA( 3 i, tan — tan 
then 
It is needless to remark that Jacobi’s transformation does not enable us to reduce 
the integral of the circular form. The difficulty which we here encounter, is exactly 
analogous to that which presents itself in the reduction of the cubic equation of ordinary 
algebra. In fact, if we were to apply Jacobi’s transformation to one only or the 
auxihary arcs just mentioned would give values of n of the fern ,+nV -H, i^e 
difficulty would depend upon the interpretation of F()i+oV^)- 
* See T'onctions Elliptiques, vol. i. p. 71. sect. 53. 
