500 
Proceedings of the Royal Society 
or 
(k'x - A y) 2 = A A' (B' - B) (ABy - A'B'a;) . 
This, again, is the equation of a parabola, which passes, like the 
others, through the origin, hut whose axis is no longer vertical. 
The converse suggests another easy but interesting problem. 
If we write £ for , rj for , and / and /' for the halves of B 
and B', we easily see that the last equation above becomes 
(i " V ) 2 = 
Every parabola passing through the origin may have its equation 
put in this form. Hence, as f and rj 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 i and y separately. A well known transformation 
leads to 
- jr-~-v = ±c/ -/) • 
whence 
Jr~- l = ±(r -/I 
Jf 2 - V= = fc ( T 
where t is some function of time or of temperature. These give 
f . = T (2/ - t) , 
V = T ( 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 be 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. 
