50 
]ME. W. SPOTTISWOODE ON THE CONTACT OF CUKVES. 
8D(™.)HDH+6 6 ^ «,H(a^+ B ^ +(g 
= 3(»Dw+®D»)(»^-«.^)H+6«(D«^-Dt,^)H 
= ( + 3i’2(;D v — 6 wD?;)H ^ — ( SviuBw + 3 w^D y — 6y wDw)H ^ 
= 3 Hy(yDw — ?yDy) ^ — 3 Hw(wDy — vDw) 
Hence, excepting the common factor (?i— 1) V, the total expression for /* 
-^3h(d>H+;^(sI ^+B Ha) -4(DHf}. 
And comparing this with Mr. Cayley’s expressions in arts. 14 and 15 of his memoir 
“ On the Conic of Five-pointic Contact,” in the Philosophical Transactions for 1859, 
pp. 376, 377 ; and bearing in mind that in his formulse we must make A=l, fjj=0, v=0, 
in order to institute a comparison; and lastly, dividing throughout by 1)~V9H^, 
and multiplying by 2, the expression above written becomes 
2J. 
di/dz 3 H y dz ' dz dy J dy dz' 
which is in fact the coefficient of YZ in his general formula, viz. 
(x|+Y|+z|)u-(|i(x^+Yf+Z^)+A(x^+Y^+Z^)) 
X X 
bU 
BU 
dx dy dz } 
The identity of the expressions for/*, as deduced by the present method, with that 
deduced by Mr. Cayley having been thus demonstrated, it is unnecessary to pursue the 
calculations further. 
The following are the principal subsidiary formulae used in the foregoing calculations. 
'h—ax-\-^y-\-yz 
>BU . .BU ^ .BU 
yi— 2^=5^, 2X-»=S-^, 
yni— 2nfe.=SO^, 2DX— apn»=Sa^, yOX = SO-^ 
dz 
