1888. ] of Curves and of Surfaces. 231 
is independent of the ratio dy: dx. The same will be true of 
the expression 
h,d& + th,dn 
err aes da+idy ~ 
Writing ¢ for p+7q, we will seek to determine its form so that 
the expression t (h,d&+th,dn) may be a perfect differential. The 
condition for this is 
OF pas 5 0 
an (th,) =4 ae (th,), 
; ot ot Chanol 
1.€. hae —ih ett gS - izgl= 0. 
This equation will enable us to obtain a suitable form for ¢, and its 
solution will involve an arbitrary function. Further, if we write 
t (h,d& + th,dn) = dw, we have 
T= (p+ig) th, cE ih, Ul = (pig) th, 5 —ingseh, 
where z=x%+1y. Also it is clear that w will a a function of z. 
Thus, if we write w= +i, we have 
Cy Os GO OS 
dz dw “On Oy oy 
Comparing these two sets of equations, we obtain 
aE oy, 21 oy, Oy oy, OF %_™ 
phy a Ox — gh, Ox = ph, oy <P 2 oy On oy 
and qh, + ph, a! mele On 0& Ou __ On 
These equations may also be written in the form 
Oe On On Om Om On 
hy = Oe Se cee ae es Oe 
Ca on OX 1 Onma Om On 
ae a ae Cen ay 
where m= p/(p’+q’) and n=q/(p*+q’). From these latter forms 
we deduce 
h?h, = (m* +n’) \(q) + Gi = = ie) + el : 
Thus if we write 
2 2 4) 2 rs) 2 
i= (52) + (Ge) = Ge) + Gq) 
and k’ = p* + q’, we have h=h,h,k. 
and 
