65 
dz dy 
Ry = Ry | Re = ede, 
Without causing confusion we can omit for two variables the 
root exponent, and when repeating the operation we can write 
Ean a 
For the rational defined in $6 the above mentioned quantity is 
constant, just as the differential coefficient of the logarithms is. 
§ 17. General rules for the rationalising are easy to fix; thus 
VRuv= Ru. Rv; 
Blur — (ub Ruyelu, Re) 7} Rule Siu, Ru); (w, Ao) | 2); 
u-bv 
VAR(ut+y) =v (Ru). (Roy; 
Re ei Ry, 
Then the following rational radices often appear: 
VR (2) == tz); 
VR Le MEN 
VR a — (en)? 
UR sra =ere ; -Rere =(sra)—; 
URire ee (Od 
Rian re = ee. (2) 5 
where we are reminded of the meaning of ¢r, mentioned in § 8; 
tan—'r represents here the opposite. 
We mention as peculiarity that the exponential function remains 
unaltered in this operation. 
UR at = at. 
§ 18. As starting point for the development in series of the product 
we choose: 
i 
oo P: 
ete HV P(e), 
l 
which formula immediately follows out of 
o Lea 
eee — 1 je 
vg} 
In a general way we can also deduce the analogon of MAcLAUrIN’s 
series: 
o Pf! dD 
y= YI HV P(2), UR Yi); ALE oO) EENES (1) 
l 
in which the index 1 refers to the values of the function and deri- 
vatives for 1. If the ratio in which the independent variable increases 
becomes 7, af the corresponding accretion of ratio of the dependent 
5 
Proceedings Royal Acad. Amsterdam. Vol. XY. 
