78 Colorado College Studies. 



In the numerator of the second fraction the same factors 



occur, the arrangement alone being dijBPereut. It only remains, 



therefore, to evaluate the denominator which is common to 



both, and which, w^hen divided by the same quantity as the 



, , A^ .-, „ a- -\- 2ac + Br 



numerator, becomes , ^ , — -, ; or the square or — -——, — h 



{R -\- ay ^ {R -\- ay 



But we have just found that 



d' 4- lac + R' — r" ., , ., , i"' 



-r=—, ^^^ = sn'v an-u, and --=—, -r, = cn-v, 



{R-\-a)- {R -^ a)' 



hence the denominator must be the square of snh- clnhi -\- cn\ 

 and when we put for dn-ii and cii-v their values 1 — Ic'sn-u and 

 1 — sn-v, this becomes 



(1 — k' srihi sii'v)'''. 



Wheri*now, all the factors of our two formulae (^3) and (24) 

 are replaced by the equivalent expressions in terms of the func- 

 tions of u and v, the two members of each are perfect squares, 

 and by extraction of the square root we obtain 



, snu cnv dnv ± snv cnu dnu ,^„ 



sn {u ± v) = . p^, , , (27 



^ 1 — k- sn'u sti'v ^ 



. en ti en V =F sn ii sn v dn u dn v , „„ 



en (u ±v) = ^-^-7^ — o ^ (28 



^ 1 — A;- sn'u sn'v 



The extraction of the square root, both here and in the 

 previous solution oE a quadratic equation in x, yields a double- 

 valued result. But we may remove all ambiguity by considering, 

 in the first place, that these formulas cannot be affected in sign 

 by a change in the value of Jc', as by making it approach zero; 

 while secondly, for A' = 0, the foregoing equations reduce to the 

 trigonometric formulas for sin {u + v) and cos {u + v). The 

 radical axis is in that case at an infinite distance, and all the 

 circles are concentric. But the above arrangement of signs is 

 that which occurs in the special case of the trigonometric for- 

 mulas, and is therefore the right one in general. The same con- 

 clusion might of course be reached by an examination of the 

 figures, in detail. 



