33 G 
PROFESSOR A. R. FORSYTH ON 
Differentiating the first two of these with respect to y and z separately and then 
inserting the values of x, y and z as they now occur on the right-hand side of (iii) 
we have 
0 dY bF m 
P m —=A~- 
dy oy 
dZ 6F,„ 
*•*=*■£ 
p dX 6F» 
A * =A V I 
rfZ cV, 
bz 
and therefore 
AfF^F.) m „ XYdZ 
A dY dZ 
= A a A— y- 
J dy dz 
by a known theorem in determinants ; thus (iii) becomes 
-t—Bw=t^~SF p 1 • 
x —« a?—« p dX t(F ra , F a ) 
dz z) 
Now expanding in partial fractions we have 
. — s'. 
K 
X' 
(x—X L ) 
dX 
da\ 
and therefore the right-hand side becomes 
Cit 
£F p A f 
U 
X a t(F,„, F„) jX 
Kv> z ) 
considered as expanded for the factors of X alone or, including in the expansion the 
1 
term arising from -it is equal to 
Cl 2 
u SFp A p 
' USF, A p ~ 
X * C< o(F Wi , I 1 n) 
4*1 
t(F w , F n ) X 
b(y,z) 
- Kv> z ) 
wherein the 2 implies summation for all values of y and z in terms of x derived 
from the equations F„,= 0 and F„=0. Since these values are to be substituted we 
have 
