64 
ANALYTICAL RESEARCHES CONNECTED WITH 
[114 
III! 
and X, g, v are indeterminate. And considering any other section represented by a 
like equation, 
(aX' + hg + gv) x + (hX + by! +fv')y + (gX' + fg + cv')z + V — p Vw = 0, 
where 
V' 2 = aX 2 + by! 2 + cv' 2 + %fg'v' + 2gvX + 2 hXg' — K, 
it may be shown by means of the lemma previously given, that the condition of 
contact is 
aXX' + bgg + cvv' + f (gv' + gv) + g (vX' -f- v'X) + h (Xg + X'g) + K — V V'. 
Suppose that X', g, v' satisfy the equations 
V' = 0, 
hX' + by! + fv' = 0, 
gX' + fg cv' — 0, 
so that the last-mentioned section becomes x = 0; and observing that the first of 
the above equations may be transformed into 
K. 
aX 4" hg' -f- gv' — — f , 
X 
(Hr 
vi’ y =m 
it is easy to obtain A/ = V^t, g'=j^, v' — 77^. The condition of contact thus becomes 
K 
X + K = 0 j 
and taking the under sign, X = V&, so that if in the above written equation we 
establish all or any of the equations A = Vgt, g = V<H}, v = ^(&, we have the equation 
of a section touching all or the corresponding sections of the sections 
x = 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 VgJ + h V<jt3 + gX'(&)x + (h Vgt + b + /VfiD) y-\- (g + /V<I3 +c \/*&)z 
+ V 2 (Vs® - (vs® - w) (va®~ =o. 
Anticipating the use of a notation the value of which will subsequently appear, 
or putting 
f= JVia© g = h = zs/mw,