Casey — On the Biquadratic. 41 



III. Equation of Differences. 



If Xy, X.2, Xz, Xi, be the roots of (2), and v^, v^^, v^ the roots of (3), we 

 have by Euler's solution, 



Hence, if s be a root of the reqxiired equation, and a^, a^, 03 the roots 



of (4) 



Z = 4 (ai + as + 2H) + 8 y (a^ + H){az+ H) 



.'. {2-4 (0^+ 03)- 8^)^=64 {ar,az^(^a^^az)E + m\. 



Now, by equation (4) 



02+03= — a^ and a^a-z = — - — . 



Hence, making these substitutions, and putting y for a^, we get 



16/ 

 {z + Ayy - 1 QEz + • = 0. (5) 



y 



The question is now reduced to the elimination of y between (4) 

 and (5), which is easily performed, as follows. From (4) we have 



v-- — + -— =0; (6) 



•^ 4 4y ^ ^ 



and eliminating in succession 



— and y''- from (5) and (6), 



we get the two quadratics, 



48/ _ %%y _ (g2 _ 16^2 + le/^) = (7) 



%%if + (s^ - 1 Qm + 4/2) 3/ + 1 2/3 = 0. (8) 



Again, eliminating y^ from (7) and (8) we get 



z^-16-g2^ + 167.2+ 72/i , 



^~" I422-96i/3 + 24/2 ' ^' 



and substituting this value of y in (7) we get the required equation — 

 s« - 48^2^ + 8 (/2 + 96^^)2" 



- 32 (32 G^ + 4:8lzK+ A5I^)?^ 



- 18 (7/2^ - 3847, m + 28%IzE:y 



- 384 {6I.'B:+ 4I.JI+ 5IJz)^ 



+ 256(1^' -271,')= 0. 



In the preparation of this Paper, Professor Ball's Memoir on the 

 Solution of the Biquadratic, published in Volume yii. of the " Quarterly 

 Journal of Pure and Applied Mathematics," has been of much use to me. 



R. I. A. TROC, SER. II., VOL. 11., .SCIEXCE. G 



