214 On a particular case of Elliptic Integrals. 



Comparing this expression with (3), A + B = 2/, A— B=0jOr 

 A =7, B=7. ^^Tience integrating (4), we get 



_./» # r d4> 



' -U \l-i{i +j V - l)sin-2</,] ^/ 1 _i2sin2<^ V \\-i[i-jV -\)^m''4>-]^^\- 



Hence a, the arc of a spherical parabola, may be represented by 

 the sum of two elhptic integrals whose parameters are imaginary. 



Now if we multiply together the imaginary parameters 

 {? + ij \^^\) and (jr—ij \^'^), their product is i^, or the para- 

 meters are reciprocal. 



It is very remarkable, that although the spherical parabola is 

 a spherical' conic, the imaginary parameters satisfy the criterion 

 of conjugation which belongs to the logarithmic form, an d not 

 that ofthe circular form. ljQim=i{i-j ^^ — \),n=i{i-\rj '^ —\). 

 These values of m and n satisfy the equation of logarithmic con- 

 jugation m + n — mn=i^, and not m — n + mn=i'^, the condition 

 of circular conjugation. 



The arc of a circle may in like manner be represented as the 

 sum of two imaginary integrals of the logarithmic form, or by 

 an imaginary arc of a parabola. 



Let x= tan 0. Then 6= /"— ^- Hence, resohdng the 

 denominator into its factors, and reducing, 



rsini 



whence 



^^=/T^.t/Trfe' • • • '^> 



TT _ P^ dx r^ dx 



2~Jo flfrV^^'t/o 1- V^^x 



(8) 



Multiply (7) by V — 1; then integrating, 



26 ^/^ l{\ + V'^x)-l{\ - \^'^x), 



or _ _ _ ^ . (9) 



4^-/_l = /(l+ K^-lxf-lil- \/-lx) 



Let arc=l, then 0=^, whence (9) may be reduced to 

 4 



7rA/~i=log(-l); 



77" , 



a result we may otherwise obtain. As also — \/ — 1 =log V —\. 



We see, therefore, that the presence of the imaginary symbol 

 A^ — 1 indicates a transition from parabolic to circular arcs, or 

 conversely. In a paper published in the Philosophical Transac- 

 tions for 1852, it has been shown how the transition may be 



