76 Colorado College Studies. 



In solving this, it is convenient to denote the quantities 

 cr + 2ac + B- and ct'c + 2ali- -\- cR- by A and B respectively. 

 The equation is then written 



AV + 2B.r {A — 2r) + B' — 4:Ar'K' + 4:r'K' = 0; 

 and is solved without difficulty, yielding 



_ B{A— 2r- ) ± 2r V ( A — r- ) (A'K' — B -) , 



X - — • —, 3 



whence we have, for the values of sir {u + v) and civ (u + v) 

 respectively, 



B— X _ A-B + AB — 2Br- ± 2r V ( A — ?•-) jA'B' — Tb^ 

 2i? ~ 2A-i? 



and 



ie + a: A-i2 — AS + 25r- T 2r \/(A — ?>-) (A-'i?- — £-) 



2R 2A'B 



As these are to be perfect squares, it is natural to inquire 

 whether the common radical term may not be, in each numer- 

 ator, the double product of two factors whose squares make up 

 the remaining terms. To test this in the case of the first frac- 

 tion, we may put 



nr + n' = A'B -f- A5 — 2Br\ and mn = r V ( A — r- ) ( A'B' — B'), 



and then forming (m'" -f- n'-y — 4:iii-n-, we obtain a perfect 

 square, from which we have 



m- — w- = A'R + AB — 2Ar'R. 



Hence we have at once the separate values of ?/r and iv, and 



may rewrite the value of sn-(u -\- v) as follows: 



B-x_ 7-iA B-B)±2r\/{A-r-){A'B'-B')+(A-r){A B-{-B) 

 -2B- 2A^ '^"^"^ 



and proceeding in a similar manner with the other fraction, we 

 shall obtain 



R+x r\AB+B)^2r\f{A-r-){A^B^—E')+{A-r^){AB-B) 

 2B ~ 2A-B '^ 



We have now to express these fractions in terms of the func- 



