122 
PROFESSOR C. NIVEN ON THE CONDUCTION 
d*V cl-V . 1 dY 1 d~Y 1 dV 
dz* dp* p dp + p- df — f f It 
+V dV 
\ =0 or l— +9i- + f)\ =0, 
«p rfz 
in which l n are here the direction cosines of the normal to a meridian section. Let 
us now replace p and z by 
p=c sinh a sin /3, z=c cosh a cos /3 
( 11 ) 
where a and (3 are the thermometric parameters of the confocal system which includes 
the principal elliptic section of the bounding surface for which a = a 0 , and whose axes 
are therefore 2c sinh a 0 , 2c cosh a 0 . 
Since p and z are conjugate functions of a, /3, 
d~Y d-V 
77 +d/ S 2 — 
rfpv /tfpY 
da 
+ 
=c~( c °sh -a —cos -/3) —+—+ 
W_IW/> 3 1 tfe 
(/p 2 ' dz 
Mor 
■eover 
whence 
and 
dY . dV . clY 
~=c cosh a sm p , +c smh a cos p 7 , 
da dp dz 
dY . tfV . dY 
—=c sjnh a cos p— -c cosh a sm pw, 
«p f/p rfe 
c(coslr a sin- /3+sinlr a cos 2 j3)y-= cosh a sin /3-y + sinh a cos /8^, 
f/p ~ da 
cosh : a sin- /3+sinli- a cos’ /3=cosh 2 a —cos- /3. 
d/3’ 
By the help of these formulae we may transform the general equation of conduction 
into 
&++ +00th + +cot +++++++= j (cosh. 2 a cos' /3)- . (12) 
in a similar manner, the equation to be satisfied at the boundary becomes 
dY 
V=0 or else— + f)c ,+coslr a — cos 2 /3Y = 0, when a=a 0 . 
• (13) 
We must also find the space element, f/E, 
c/E =pd (b.d pdz=pd (fid ad (3 
dp dp 
dad 7/3 
dz dz 
da’ 77 
