710 



I have made use of my formulae in the reduction of one class of 

 the elliptic integral of the third kind for the purposes of tabulation. 



In its ordinary form, this function has three variables, and, as 

 Legendre justly remarks, a table of treble entry would be intolerable. 

 If, therefore, it is to be tabulated at all, the first step in the question 

 is to reduce it to a form involving two variables. By the help of 

 some theorems of Jacobi, Legendre succeeded in effecting this where 

 the elliptic function of the third kind has its parameter negative and 

 less than the square of the modulus. 



By the help of my formula, I have succeeded in reducing the case 

 where the parameter is negative and greater than unity. The steps 

 are the mere counterpart of Legendre' s work in the second supple- 

 ment of the treatise on Elliptic Functions. 



Both Legendre' s case and mine are of the logarithmic form, and 

 can therefore be reduced to one another by algebraic transformation. 

 The cases where the parameter is positive, or negative and interme- 

 diate between unity and the square of the modulus, are still unre- 

 duced. The difficulty is exactly analogous to that between the two 

 cases of cubic equations, and this analogy is even carried into the 

 very form of the solution. 



Dr. Booth's application of the trigonometry of the parabola to the 

 reducible case of the cubic equation, affords some hope that a corre- 

 lative calculus may exist, particular cases of which may solve the 

 cases now irreducible, just as the calculus of elliptic functions in- 

 cludes the trigonometry both of the parabola and the circle. My 

 own investigations on this subject are still without any useful result. 



Let 



cos 0! = cos 2 cos 3 sin 2 sin 3 V 1 sin 2 sin 2 X ; 



_r 



J (l-si 



sin 2 0sin 2 0)*' 

 E0=j" (1 -sin 2 sm 2 0)*rf0, 



r , f cos 2 Od<j> 



J (I -sin 2 sin 2 0)* cos 2 0<fy' 



(1) Ffc-F,, + F,,=0 



(2) E0 X E0 2 + E0 3 =sin 2 sin X sin 2 sin 3 



(3) Y0 1 -Y0 2 + Y0 3 = -cos 2 tan 1 tan 2 tan 3 



