STEINER’S EXTENSION OF MALFATTl’S PROBLEM. 
259 
where 
y ^ + cv^ + 2f(jtjv 4- 2gvk -j- 2 — K, 
where X, g, v are indeterminate. And considering- any other section represented by 
a like equation, 
(aV + -\-gv)x-\r (/«X' + h^/J -k-p')y + ( g^l +/^' + CJ'') z — jo v'^ = 
where 
y ' = aX'^ + 2fg'v' + 2^-^'^' + 2 Ax'iW/' — K, 
it may be shown by means of the lemma previously given, that the condition of 
contact is 
al >! + + cv'J ■\-f{gv' ■\-i/Jv)-\^g{vh!-\- v'yC) 4- h {X(jtJ + i K = w^- 
Suppose that a ', [jJ, v ' satisfy the equations 
v'=o. 
hK'-]-b[jJ-{-fv=0, 
g^'+f(^'+ci''=0, 
so that the last mentioned section becomes .r=0; and observing that the first of 
the above equations may be transformed into 
aX' + hg -\-gv'——,, 
it is easy to obtain a '=\/ 9 [, ( m '=. 
The condition of contact becomes 
V ^ V ^ 
K 
=X-1-K=0. 
— 
And taking the under sign, >i=\/9[, so that if in the above written equation we 
establish all or any of the equations X=\/^, we have the equa- 
tion of a section touching all or the corresponding sections of the sections 
a; = 0, y=0, z = 0. 
In particular we have for a solution of the problem of tactions, the following 
equation of the section touching x=0, y=0, z=0, viz. 
(a^/a+/;^/S+^^/€)J;+(7i^/3+^'^/l+/^/C)I/+(g^/a+/^/l+c^/C)^ 
+4^^/2(^/33c-JF)(^/ac-®)(^/aS-lE))1^«=o. 
Anticipating the use of a notation the value of which will subsequently appear, or 
putting 
f= f'Ws/v'm-sr, g= i'S'/ j= v'2^wm, 
values which give 
K^=-f^-g^-h^+2g^h^-l-2h^f^4-2fV-^^^"’ 
the equation of the section in question is 
f2 
/ r 9 t 
/ r*o o I I o \ 
.2 
2 
.2 
2 
fgh ^ — p K 
