the Elliptic Equation. 



203 



The proof may be found in Legendre's Fonctions Elliptiques, 

 vol. iii. p. 174. Curiously enough, the enunciation is confined 

 to the multipUcation of the functions, while the proof is perfectly 

 general. Whether its generality escaped the notice of Legendre 

 and Jacobi I am uncertain. 



It must be remarked, that when ■^q is an odd multiple of 



pr-, we have ?- = 0, and e= >,, = sin ^ : but in no other 



2' ' 1 + COS0' ' 



case. The variation of i/r^ does not yield a family of circles 



having a common centre of similitude. 



It should also be noticed that the theorem is one-sided : it 

 will not do to start from the other end B of the diameter. 



Now let us suppose the outer circle of this diagram to be a 

 circle of the stereographic projection, the distance between its 

 centre and that of the sphere being sin 6 ; then if the longi- 

 tudes of the points Mq, M„ Mg be ^q, <^i, ^2> *^^ equation 



and consequently the elliptic equation 



cos 00= cos 0, cos 02 + sill 01 sin ^^(l ~~ siu^ sin^ 0)- 

 must also hold : since in ^r-r/r—:- = A: -tttst-tt the factor k is 



A(^, 0) ^{6',^lr) 



constant,and therefore the equation F(^, 0^) — r(^, 0,) = F(^,0o) 

 is merely the equation Y{6'^\r^} — Y{ff^^) = Y{e'^^Q), with each 

 term multiplied by a constant factor. 



To fix our ideas, it will be as well to exhibit this theorem by 



a diagram. As before, let AMq=2-^q, ilMi = 2^i, AM2 = 2-«|r2. 



Then the angles AOMo=0o, AOM,=0„ AOM2 = 02; and, as 

 before, 



^.„ l-A(^'-»Ir.) - ,, , , 2cOS-»/ro 



