76 
to the purpose of tabulation than is the elementary integral 
used by Legendre and others, viz. : 
f c dd 
jov/ 1 - Fsin. 2 0 
i 
to which the former integral can be readily reduced. In 
tracing the curve whose polar equation is r 2 = 2 1 s / 1 + nCos.md 
by polar coordina/tes r, d the periodicity of cubic integrals is 
readily perceived. 
And here I must express my surprise that this method of 
tracing the cubic integral has not been made use of in 
tracing elliptic functions. That this has not been done I 
infer from the circumstance of not finding it in Cayley’s 
recent Treatise on Elliptic Functions, 
2. Let a , b, c, be positive, and taken in the order of mag- 
nitude, that is 
If the roots are not positive they can be readily made so 
by the linear transformation x=A + 0 2 . 
With a view, therefore, of fixing palpably the values of 
the limits a, (3 with respect to the roots a, b, c, it will be 
necessary to trace, by the usual rectangular coordinate 
method, the curve whose equation is 
ODEe BECA 
Let Ox, O y be rectangular coordinates origin at 0. 
When then, OR, Jabc = 1. The negative branch of 
the curve extends to infinity with a continually diminishing 
