813] 
NOTE ON MR GRIFFITHS’ PAPER. 
59' 
That is, in defining the a 0> a 2) a 4 , functions of the two arbitrary coefficients a, (3, 
as above, we have in effect so determined them that Vl — u 2 shall be = Vl — x 2 multiplied 
by a rational function of x. 
We can then further determine a 0 , a. 2 , a 4 in such wise that the change of x into 
r- shall change u into —; and, this being so, making the change in Vl — u 2 , we 
rCCC A*Zl 
obtain Vl — \ 2 u 2 in the form, Vl — k 2 x 2 multiplied by a rational function of x; viz. 
it is a function of x such that 
du _ Mdu 
VI — u 2 .1 — A 2 u 2 Vl — x 2 .1 — k 2 x 2 
The theory is thus in effect Jacobi’s—with the novelty of combining two lower 
transformations in such wise that the assumed expression for u as a rational function 
of x shall give 
Vl — w 2 = V1 —x 2 multiplied by a rational function of x. 
It is not necessary that the equations 
y = rational function of x and z = rational function of x 
should be elliptic-function transformations. All that is required is that they should 
be such as to give Vl — y 2 and VI — z 2 each =Vl— x 2 multiplied by a rational function 
of X.