499 
of Edinburgh, Session 1870 - 71 . 
The surface of the third order, represented by either of the two 
latter equations, is well known, and the property above shows a 
curious relation between certain of its vectors and those of a central 
surface of the second order. It has also interesting applications 
to the lines of curvature of the surface. 
If £ and 7 ] be unrestricted, the theorem above may be put in the 
more general form that the two following equations are conse- 
quences one of the other, viz. : — 
£ V.rjtpy 
$ 3 * -£ ( P£ ( P 2 £ $ 3 .rj(pr](p 2 7] 
r) __ I • £$£ 
.rjpr)(p 2 r] S* .£<p£ty 2 £ 
From either of them we obtain the equation 
S<p£<P0 = S 5 ,£(p£^> 2 £ S 5 -rtf rtf 2 r] , 
which, taken along with one of the others, gives a singular theorem 
when translated into ordinary algebra. 
2. Relation between corresponding Ordinates of two Parabolas. 
Two projectiles are anyhow projected simultaneously from a 
point, what is the relation between their vertical heights at any 
instant ? 
This simple inquiry, which was instituted in consequence of some 
results recently obtained from thermo-electric experiments (see ante . 
p. 311) carried on at high temperatures, where the indications given 
by two separate circuits, immersed in the same hot and cold bodies, 
were used as ordinate and abscissa, leads to a very curious conse- 
quence. 
Let 
x = At (B - t) 
and 
y = A7(B' - t) 
be any two parabolas whose axes are vertical, and which pass 
through the origin. We have 
A'x — Ay [- ^ A'x — Ay q 
•" ii - i; -l. ' aa i ii is J' 
