500 
Proceedings of the Royal Society 
or 
(A'x - Ay) 2 = A A' (B' - B) (AB y - A'BV) . 
This, again, is the equation of a parabola, which passes, like the 
others, through the origin, but whose axis is no longer vertical. 
The converse suggests another easy but interesting problem. 
If we write £ for 
x 
A 
V 
for — , 
A' 
and / and f for the halves of B 
and B', we easily see that the last equation above becomes 
if-f'U'l-fn)- 
Every parabola passing through the origin may have its equation 
put in this form. Hence, as | and y are dependent on one another 
(in the thermo-electric as in the projectile case) only as being 
both functions of temperature, or of time, it is obvious that we must 
seek to break this expression up into a linear relation between 
functions of £ and y separately. A well known transformation 
leads to 
JT^i- Jr ->/ = ± (/-/r 
whence 
jr -% = ±(t -/). 
JJ 2 - 1J = ± (t - /') , 
where r is some function of time or of temperature. These give 
i = T (2 f - r) , 
V = r (2/ - r ) • 
Hence, in the thermo-electric case, if we obtain a parabola by using, 
as ordinate and abscissa, the simultaneous indications of any two 
circuits whose junctions are at the same temperatures, and if one of 
them gives a parabola (with axis vertical) in terms of absolute 
temperature, r must he a linear function of the difference of absolute 
temperatures of the junctions, and, therefore, the other circuit gives 
a similarly situated parabola in terms of the absolute tempera¬ 
ture. 
