1S81.] of Cartesians and other Quartics. 85 



and therefore the quartic curves derived from 



x + iy = sn (£ + irj) 



may be also expressed by 



(1 - h)r i - (1 + A^dn 2f = 2r 2 en 2f ) 

 (1 - k)i\ + (1 + h)r l dn liv' = 2r 2 en 2irjj ' 



where £' + i'?/ = | (1 -f A;) (| + iv) '> 



with similar expressions connecting r x , r 2 , r 3 ; r t , r 3 , r 4 and r 2 , r s , r 4 . 



The curves obtained from x + iy = dn (£ + ^) may be similarly 

 expressed. 



8. The integral 



dz 



h 



*/{(z-m) 2 + n 2 .{z-m'y + ri 2 }' 

 written in the form 



dz 



/; 



\/(z — m — in.z — m + in . z — m'+ in. z — m — in) ' 

 when considered as a case of 



dz 



/. 



sj{z — a . z — b . z — c . z — d)' 



by putting a — m + in, 



b — m — in, 



c = m' — in, 



d = m f + in, 



will give ultimately in a real form the substitutions required for the 

 reduction of the integral in general, and also the quartic curves 

 whose foci A, B, C, D are given by 



For tf = 



z = m + in, m — in, m'—iri, m'+in. 



b — c a — d 

 a — c b — d 



m — m — i . n — n m — m'+i .n — n' 

 m — m + i . n + n m — m — i.n +n' 

 (m - vif +(n- n 'f _ A l>' 

 ' (m - m'f + (n + n'f ~ AC ' 



