PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 115 
But if we admit that at one point of space there may be a particle of matter 
of mass m, we have, of course, 
m2 
uU = Tp? 5 
so that 
= 2p 
Vai = oe inte 
which gives as a particular integral 
Una = Up. 
Hence, in this case, 
2 
d= = ugdpg-! = — 72 (2Up SUpdp + dp) 
mi i 2pdTp _ dp 
+ TA (2pSpdp + dpp’) = — m? ae Tp?) 
or 
= ips 
The corresponding surfaces are the electric images of the confocal quadrics, 
taken from the common centre, and include Fresnel’s surface of elasticity. 
13. It follows, from what we have just proved, that the only orthogonal 
surfaces which divide all space into indefinitely small cubes are planes and their 
electric images, or images of images, &c. These are all, therefore, included in 
a triple series of spheres having a common point, and their centres in three 
rectangular axes passing through that point. 
In fact, if in (7) we put for - 
ow = (- + Th cs 
we have 
dc’ 
Fe os Co &e., 
whence 
de OS a derde, 
Sa dy Sa. Te &C., 
and 
do do~’ do- wu 
Ue Ge ee 
Hence the electric image of any orthogonal system is also orthogonal ; and, if 
the system cut space into cubes, so does the image. 
14. We are now prepared to introduce the conditions that the surfaces 
(1) shall each be isothermal. Ifh, ,, h, represent their temperatures, these 
conditions are simply 
Vi 0, Vit=0, V2, = 0.. : : (22). 
To express these in another form we must now differentiate equations (2). 
15. By (2) we have 
VOL. XXVII. PART I. 2G 

