135 
Then, by the theory of equations 
r + S -> rt - 
2a 
— — ^ a wr 
:,x^-^ny/r.x= - \// 3 -\/v = 9 -^± 
2 2 
_\2 
2?i y' 'p 
Tirr _ 
= ^ I I - -x/r - - v/ 6=F v/ « j 
Similarly 
»=§ {v'j'-Hv'sTv'q ^^Vr- V~s^V~t\ 
These are, therefore, the four roots of 
x^ + an^Jtz^x + c = 0 
8. There are other methods of solving biquadratics 
depending upon general views applicable to equations of all 
dimensions — 
Tschirnhausen’s method is to reduce 
x^ + 7io(^ + ao(^ + bx + c = 0 (33) 
to j/^ + A 2/^ + B = 0 
by eleminating x between (83) and the following 
oi^ + ax + p + y-O (34) 
where a, (3, are indeterminate constants. 
Lagrange has applied the theory of symmetrical functions 
to the solution of equations of the third and fourth degrees. 
The above methods effect the object in question, but, by 
processes far more complex than those adopted in this paper. 
Professor Young has given a very ingenious solution of a 
biquadralic. (Young’s Equations, page 454.) 
“ The Castel Nuovo Lignite Deposit, near San Giovanni 
Tuscany,” by Watson Smith, F.C.S., F.I.C. 
During the Easter of 1877, I had been requested by Jas. 
Young, Esq., LL.D., F.KS., to execute a commission in South 
Italy for him, and was on my return to him in Florence, 
when on passing a small station, “San Giovanni,” still some 
